#descriptive statistics

Research Methods

In some ways, Psychology is a science. It is difficult to define it as a science since some approaches cannot depend on the scientific method to perform experiments due to the unpredictable nature of the human psyche. While in Biology, it is easy to isolate two variables to study, however, in Psychology some complications such as hindsight bias can appear. Hindsight bias is the tendency to assume that you know the results all along. 

There are several ways psychologists can use psychology to change the world. Some conduct research to solve practical problems. This is called applied research. Others may conduct basic research, which is when psychologists conduct experiments not intended to have immediate real world effects.

Terminology

Most psychological research is guided by a hypothesis. It expresses a relationship between two variables, which are things that can vary among participants in an experiment. Skin colour, age and hair texture are all variables. The dependent variable is the variable who depends on the independent variable (hence the name.) The independent variable is typically the variable being altered in some way, which impacts the dependent variable which is typically the variable being studied. For example, if you wanted to look and see if the heat of the environment impacts the structure of a protein, the dependent variable is the protein structure, and the independent variable is the temperature. A theory attempts to explain certain phenomena, and gives scientists the ability to generate testable hypotheses. Psychologists also have to provide an operational definition of the variable that they will study. This means that they will explain how they measure that variable. 

Good research is valid and reliable. To be reliable means that the research can be replicated, and to be valid means it measures what the researcher wants it to measure. For example, if someone made Albert Einstein take an IQ test which rated his intelligence from 1 to 100 to find out if he was above or below average intelligence if he were to get a 9 consistently, the test is clearly reliable but invalid. If Albert Einstein were to take this test and get a 100 then a 9, then this test would be unreliable and invalid.

Before a hypothesis is investigated, who or what is being studied needs to be decided. Participants are the individuals who the research is conducted on, and the process that participants are chosen is called sampling. The goal of sampling is to pick a representative group to allow for generalisations to be made. Random selection is ideal as every member of the community has an equal chance to be chosen. (In a way that does not allow for bias so that the selection is truly random). Stratified Sampling is another process that makes sure a sample represents a population. For example, if in my study I am including two religious groups: Religion A and Religion B, and I feel that religion may have some bearing on my research, I could divide the group of possible participants into two groups and then randomly choose people, thus ensuring representation from two groups which may affect each other.

Experimental Method

There are two kinds of experiments: laboratory experiments and field experiments. Laboratory Experiments are done in a lab (hence the name) which is an extremely controlled environment. Field Experiments are performed in the real world. Laboratory Experiments allow for more control, while Field Experiments allow for more realism. Experiments are the preferred method of research among psychologists because it is the best way to show a casual relationship. It allows manipulation of an independent variable and to control for confounding variables which is any difference between the experimental and control conditions except for the indpendent variable which can cause changes in the dependent variable. If I want to prove that cheese causes nightmares, it’s best to perform an experiment where I make sure that it is the cheese leading to the nightmares, and not other factors such as my subjects being sleep deprived, or being forcefed ounces of cheese which would inevitably cause nightmares.

Assignment is the process by which participants are put into a group (either experimental or control.) Random assignment means that each participant has an equal chance of being placed into any group. Random assignment reduces the effect of participant-relevant confounding variables. If in my cheese experiment, I gave my participants the option to choose whether they would like to be the control or experimental group, people who don’t like cheese would be more likely to avoid the experimental group, meaning the people eating massive amounts of cheese likely wouldn’t be scarred by eating ounces of it. Group matching can be used to make sure experimental and control groups are equivelant on some criterion. For example, if I want to make sure I have equal amounts of men and women in my groups, I could split my group into men and women, and then assign them to control or experiment. 

Situation-relevant confounding variables can also prove to be an issue. The situations that the different groups are put into must be equivelant except for the independent variable. These can include the weather, the ambiance, or the lighting. Experimenter Bias is a specific type of situation-relevant confounding variable. It is the tendency for the experimenter to treat members of the control or experimental group differently to increase their chances of confirming their hypothesis. (It isn’t a conscious action). Double-blind procedures can eliminate this bias. Normally, the way these work is that a researcher has someone who doesn’t know who is control and who is experimental interact with the subjects. Single-Blind procedures are when only the subjects don’t know which group they are assigned to. This minimises demand characteristics and types of response and participant bias. Demand characteristics are cues about what the study’s purpose is. Participants may try to use these to figure out how they are supposed to behave. One type of response bias is social desirability which is the tendency to give answers which will reflect best on the person’s character.

Control groups give something to compare the results of the experimental group to, thus concluding whether there is a relationship between the two variables. In fact, simply selecting a group of people to experiment on can affect the performance of the group regardless of what is done. This is what is called the Hawthorne Effect.

Another method of control is called the placebo method. This is typically used in drug experiments. The control group are given a sugar pill- that looks identical to the functioning pill. This allows psychologists to seperate the physiological effects of the pill to the psychological effects known as the placebo effect.

In some cases, participants can be used as their own control group- this is known as counterbalancing. This can be flawed as it can cause order effects. For example, if a psychologist was seeing how cheese affects peoples math skills, they can give the participants a math test, feed them cheese, and then give them a second math test, however they may do better on the second test simply because they’re doing it over again. Counterbalancing helps deal with this problem as half of the group gets the cheese first, and the other half doesn’t.

Correlational Method

Correlation expressed a relationship between two variables. It does not imply causation. 

Click here: https://twentytwowords.com/funny-graphs-show-correlation-between-completely-unrelated-stats-9-pictures/ to see some very bizarre examples of this very important rule when using the correlational method. A positive correlation between two things means that they have a direct relationship (If I do more of Action A, I will get more of Object B) if they have a negative correlation, it implies an indirect relationship (If I do more of A, I will get less of B). 

Some hypotheses are impossible to make into experiments, which is when the correlational method can come in handy. These are situations where the independent variable is impossible to control; for example if I want to behave differences in men and women’s behaviour, I can’t control the independent variable because it will be predetermined, meaning I can’t guarantee differences in behaviour can be ascribed to gender differences or other phenomena I can’t account for with random assignment. If I try to control all other parts of the research in this case, it is considered an ex post facto study. 

Another popular use of the correlational method is the survey method. For example, If I want to see if wearing socks leads to depression, I can ask people to fill out a survey detailing if they have depression and how often they wear socks. This method, like any correlational method cannot identify a cause-effect relationship, only establish a relationship. (I don’t know how many ways I can say this. Stop using correlational studies to prove your bogus cause-effect relationship hypotheses. Vaccines don’t cause autism.) There is no independent or dependent variable in a survey. One major flaw with the survey method is that there is no way to control participant-relevant confounding variables. They can be massively convenient, as a psychologist can have several people mail back responses, however, a random sample is difficult to obtain because few people will send it back (it has a low response rate)

Naturalistic Observation

Naturalistic observation is unobtrusive. It provides no control, however gives the researcher a detailed look into the behaviour of a person. One example is the case study method where researchers spend years studying one, or a small group of people, giving intimate details into how their lives are and how certain events may lead to a certain behaviour they want to observe. These types of studies cannot be generalised due to their small sample size.

Statistics

Now that you have all your data, you need to be able to interpret it, otherwise; what’s the point of having all that data? Descriptive statistics is a creatively named part of statistics where stats are described. You can summarize your data with a frequency distribution. Take an example where you were observing what colour everyone in the office wore one day. A frequency distribution would tell you how many people wore red, how many wore black etc etc. You can graph this data and turn it into line graphs called frequency polygons or bar graphs called histograms. Frequency always goes on the y axis.

Another way to summarize a set of data is by looking at the central tendency. We can do this by looking at the mean (adding all the numbers together and dividing by the number of scores) the median (write all the numbers in ascending orders, then find the middle number, or if there is an even amount of numbers, the average of the two middle numbers) and the mode (the most frequently appearing number). Some sets of data have more than one mode. A bimodal has 2 scores appear equally frequently and more frequently than the other scores.

The mean, while the most commonly used can be affected by outliers. Outliers are scores that are way outside of the norm. For example, take GPA, which is the average of all of your grades. Say for 4 years you were a straight A student in all 6 of your classes, except one year, you flub and get an F in math. Your GPA would be a 3.33, implying that you are a B average student. Your GPA was negatively skewed by that one bad year in math, meaning that the skew was caused by an extremely low score. If you were a straight D student, but one year managed to get an A in both Math and English, your GPA would be a 2.00, implying you are a C average student. This data is said to be positively skewed, as an abnormally high score caused the skew. In unscrewed data, the median and mean are exactly the same. With negatively skewed data, on a graph, the mean shift left, while with positively skewed data, the mean shifts right. 

Measures of variability can also describe the data you have collected. These include range, variance, and standard deviation. These depict the diversity of the data. Standard deviation is the square root of the variance. They relate the average distance of the scores from the mean. A higher variance and SD implies more spread out data. The range is the difference between the highest and lowest score. If you want to compare scores from different distributions, you can convert the scores into z scores. These measure the distance of a score from the mean in units of standard deviation. Say a student got a 72 on a test where the average score was an 80 with a SD of 8. Her z score would be -1. If someone else got 84 on the same test, her z score would be +0.5. The normal curve is a bell shaped curve, where the area under the curve lying between two z scores has been predetermined. Z scores calculate the distance of a score from the mean, while percentiles indicate the distance of a score from 0. Someone who scores a 1250 on their SAT is around the 81st percentile, meaning they scores Better than 81% of test takers. Someone in the 50th percentile has a z score of 0. Someone with a z score of 98 has a z score of 2. See the pattern?

Correlations

Correlations observe the relationships between two variables. They can either be strong or weak; this can be calculated with a correlation coefficient. These can range from -1 to +1, with -1 being a strong negative correlation and +1 being a strong positive correlation. 0 means that there is no correlation. Correlation can be graphed with a scatter plot. If there is a strong correlation, the dots on the scatter plot will come close to a straight line. The line of best fit (also known as the regression line) is a line drawn through the plot that minimizes the distance of all plots from the line.

Inferential Statistics

Inferential Statistics determine whether findings can be generalised to a larger population. It is impossible to guarantee that a sample is representative of the entire population. The extent of the difference between the sample and population is called sampling error. There are many tests that determine if findings can be applied to a larger sample of people such as chi square, t-tests, and ANOVAs. These tests yield the p value which observes the probability that the results were due to random chance. A p value of .05 is the cut off for statistically significant results. A p value of .05 means that there is a 5% chance that the results were due to random chance.

APA Ethical Guidelines

Through its history, Psychology has lead to many horrific experiments which have caused significant harm to its patients. One example is the Little Albert experiment, done by John B Watson. He clanged an iron pole every time the little boy reached out to touch a white rat. Even without the sound, the boy was afraid of the rat showing emotions could be conditioned. While this research provided important information for scientists, had Albert not died at 6, could have had serious implications for him in his life outside of the experiment. The APA now has a series of guidelines to assure research does not cause serious harm. Before research can begin, the study must be presented to the Institutional Review Board (IRB) which reviews proposals for ethical violations and procedural errors. The regulations vary based on whether the subject is a human or an animal.

For animal research, the study must meet these requirements:

- They must have a clear scientific purpose

- The research must answer a specific and important scientific question

- Animals chosen must be best suited to answer said question

- The animals must be cared for and housed humanely

- The animal subjects must be acquired legally and purchased from accredited companies. Wild animals must be caught humanely

- Experimental design must employ the least amount of suffering possible. 

Research involving human subjects must meet these requirements: 

- No coercion. Participation must be voluntary

-Informed Consent: Participants must know that they are involved in research. If deception is necessary, the research they consent to should be similar enough to the research they participate in.

-Anonymity/Confidentiality: Participants have anonymity when no data is collected that can allow a persons response to be matched with their name. If anonymity is not possible, confidentiality is necessary, where the researcher doesn’t identify the source of their data.

-Risk: Participants can’t be placed at any significant mental or physical risk.

10 Types of Data Analysis

(Mathematical and Statistical Methods) 1. Descriptive Analysis: It does what the name suggests - "Describe." It looks at data and analyzes past events for deciding how to approach the future.

2.  Regression Analysis: It allows modeling the relationship between a dependent variable and one or more independent variables.

3.  Dispersion Analysis: Dispersion is the spread to which a set of data is stretched. It is a technique of describing how extended a set of data is.

4.  Factor Analysis: Factor analysis is a regression based data analysis technique, used to find an underlying structure in a set of variables.

5.  Discriminant Analysis: The discriminant analysis utilizes variable measurements on different groups of items to underline points that distinguish the groups.

6.  Time Series: It is a process of modeling and explaining time-dependent series of data points. The goal is to draw meaningful information (rules, patterns) from the shape of data.  

(Methods Based on The Artificial Intelligence, Machine Learning and Heuristic Algorithms)

7.  Neural Networks: They present a brain metaphor for information processing. These models are biologically inspired computational models. They consist of an interconnected group of artificial neurons and process information using computation approach.

8.  Decision Trees: The decision tree is a tree-shaped diagram that represents classification or regression models. it divides a data set into smaller and smaller sub data sets while at the same time a related decision tree is continuously developed.

9.  Evolutionary Algorithms: A common concept that combines many different types of data analysis using evolutionary algorithms. Most popular of them are: genetic algorithms, genetic programming, and co-evolutionary algorithms.

Día 12 de 100 

La primera foto es de esta mañana, he estado haciendo resúmenes/esquemas del tema 1 de historia y estoy haciendo malabares entre un libro antiguo de bachillerato y uno de historia contemporánea de Artola. 

Por la tarde he seguido (ahora mismo son casi las 8 de la tarde) y ya he parado con historia. Al final he hecho 2 partes de 5 que engloban el tema 1 de la asignatura. Voy a ponerme ahora con la teoría de análisis descriptivo de datos en sociología y si me da tiempo y no estoy muy cansada leeré historia antes de dormir. 

Espero poder acabar los resúmenes del siglo XIX esta semana e incluso empezar el siglo XX. ¿He dicho ya que odio historia? ¡Qué mal lo paso con esta asignatura! Se me da fatal y me hace sentir muy tonta, además de que luego en el examen no se ni lo que ponerle. 

Ayer (día 11 de 100) fue un día bastante flojo. Estuve todo el día fuera del piso, por la mañana no estudié ni hice nada y por la tarde estuvimos haciendo unos compañeros y yo ejercicios de análisis. Pero por la noche leí unos temas de historia por encima antes de dormir. 

Analysis of data that helps describe, show or summarise data

Patterns may emerge from data

Does NOT allow us to make conclusions beyond data we have analysed

Does NOT allow us to make conclusions regarding any hypotheses

Simply way to describe data

Distribution (i.e. percentages, histograms) - shape can be described via skew

Central tendency (3 major types: mean, median, mode)

The analysis of variance F test, or ANOVA, is used to compare the means of I populations, where the following hypotheses are tested:

H₀: μ₁ = μ₂ = ... = μI vs. Hₐ: not all population means are equal

Note that the alternative hypothesis is not Hₐ: μ₁ ≠ μ₂ ≠ ... ≠ μᵢ. It is not needed for all the population means to differ from one another in order to reject the null hypothesis, only one population mean is required to differ from the other means in order to reject the null hypothesis.

There are many possibilities for the null hypothesis to be false. Given four population means from four different populations, perhaps μ₁ = μ₃ = μ₄, yet μ₂ differs. Perhaps μ₂ = μ₃, yet μ₁ and μ₄ differ. Perhaps none of the means are equal. Therefore, the alternative hypothesis is no longer one-sided or two-sided, but "many-sided."

To understand the idea of ANOVA, consider the following two graphs:

The sample means in (A) are the same for (B), yet the variability for the samples in (B) is smaller than (A)'s variability.

If another sample from each population in (A) was taken, the sample means may differ from the original sample means in (A) due to the large variability in (A). Therefore, the observed difference in sample means can be due to the chance of variation, and the researcher may not reject the null hypothesis of all four population means being equal.

However, with (B), the variability is lower and so if another sample from each population in (B) was taken, it is unlikely the sample means would differ from the original samples of (B). It would be unlikely for the new sample mean for Population 1 to be as low as the new sample mean for Population 4.

Therefore, it appears the difference in sample means in (B) is not only due to the chance of variation but it can also be due to a true difference in population means. Even though the sample means are the same in (A) and (B), the researcher would likely not reject the null hypothesis for (A) but reject the null hypothesis for (B) because of (B)'s lower variability.

Although the above explanation gives an idea of ANOVA, it ignores the importance of sample size.

Assumptions for ANOVA The following are three assumptions when conducting inference procedures using ANOVA:

1. The variable of interest follows a normal distribution for each population of interest. 2. Each population's standard deviation are assumed to be equal, and so there is a common standard deviation σ. 3. Each of the I samples is a simple random sample from the respective populations. In the case of an experiment, then the treatments are randomly assigned to the individuals.

Analysis of Variance Analysis of variance compares the variation between the sample means to the variation within the individual samples.

If all of the population means are equal, then all of the observations come from the same normal distribution. It is expected that both the variation between the sample means and the variation within the individual samples represent only the inherent variability of the variable, which is the common standard deviation σ. Therefore, there should not be much difference between these two variations. However, if the means differ a lot, then the variation between the sample means is much larger than the variation within the individual samples.

Test Statistic F and F Distributions The ANOVA test statistic F for testing the equality of several population means is of the following form:

Like the t distributions, there are many F distributions. While the t distributions depend on only one degrees of freedom, the F distributions depend on two different degrees of freedom. One for the numerator of F and another one for the denominator of F.

The F distributions are all skewed to the right and never negative, since the F test statistic is a ratio between two variations, and variations are never negative.

The higher the value of the F test statistic, the greater the difference is between the sample means, which gives more evidence against the null hypothesis.

Example To find the upper 0.05 critical value for the F4,19 distribution, where 4 is the numerator degrees of freedom, or df₁, and 19 is the denominator degrees of freedom, or df₂, find df₁ = 4 and df₂ = 19 on the F* table, then the row for 0.05 is the critical value F* = 2.90.

Suppose there is a simple random sample of size nᵢ from each I different, normally distributed populations with a common standard deviation σ. Then the total sample size is the following:

N = n₁ + n₂ + ... + nI

For each I population, the I sample means and I sample standard deviations are calculated:

The researcher would like to test the following:

H₀: μ₁ = μ₂ = ... = μI vs. Hₐ: not all population means are equal

The overall mean x̿ for all samples is then the following:

Example A researcher would like to compare the effective of three different weight loss programs. Twenty overweight adults volunteer to participate in an experiment. Seven of the adults are randomly assigned to Program 1, seven are randomly assigned to Program 2, and six are randomly assigned to Program 3. The weight loss in pounds of each subject is measured at the end of three months. The researcher conducts a formal hypothesis test using a 5% level of significance: H₀: μ₁ = μ₂ = μ₃ vs. Hₐ: not all of the population means are equal Therefore, there are three populations. The first sample n₁ = 7 is Population 1, which are the overweight adults who follow Diet Program 1 for three months. The second sample n₂ = 7 is Population 2, which are the overweight adults who follow Diet Program 2 for three months. The last sample n₃ = 6 is Population 3, which are the overweight adults who follow Diet Program 3 for three months. The following is the data obtained after three months:

Using a program, each of the samples' box-plots are examined:

It is difficult to assess the normality of the distributions from box-plots based on small sample sizes. For the hypothesis test to be accurate, it must be assumed the weight losses follow a normal distribution for each of the three programs. According to the box-plots, it does not appear the three population means for the three programs are equal, but the hypothesis test will determine if the observed difference in these sample means is significant. From the data, the following is calculated: n₁ = 7, x̄₁ = 9.14, s₁ = 2.91 n₂ = 7, x̄₂ = 5.57, s₂ = 2.51 n₃ = 6, x̄₃ = 11.17, s₃ = 2.86 There are a total of I = 3 groups, and so the total sample size is N = 7 + 7 + 6 = 20. The overall mean x̿ is calculated: Σnᵢx̄ᵢ = (7)(9.14) + (7)(5.57) + (6)(11.17) = 169.99 x̿ = 169.99/N = 169.99/20 ≈ 8.50

Numerator and Denominator of the F Test Statistic The numerator of the F test statistic measures the variation between the sample means. The variation between the sample means is measured using the sum of square for groups, denoted as SSG, where SSG is the following formula:

Then the mean square for groups, or MSG, is the following:

MSG = SSG/(I – 1)

MSG gives a measure of how far apart the I sample means are to each other, and so MSG is the numerator of the F test statistic with I – 1 degrees of freedom.

Example For the previous example, SSG = (7)(9.14 – 8.50)² + (7)(5.57 – 8.50)² + (6)(11.17 – 8.50)² ≈ 105.73. MSG = SSG/(I – 1) = 105.73/2 ≈ 52.87

The denominator of the F test statistic measures the variation within the individual samples. The variation within individual samples is measured using the sum of squares for error, or SSE, where SSE is the following formula:

Then the mean square for error, or MSE, is the following:

MSE = SSE/(N – I)

MSE is the denominator of the F test statistic with N – I degrees of freedom.

Example For the previous example, SSE = (7 – 1)(2.91)² + (7 – 1)(2.51)² + (6 – 1)(2.86)² ≈ 129.51. MSE = SSE/(N – I) = 129.51/17 ≈ 7.62

Notice that MSE is an extension of the pooled sample variance in the two-sample case. Therefore, a good estimate for the pooled sample variance sp² is MSE and a good estimate for the common population standard deviation σ is sp = √(MSE).

Therefore, the F test statistic is formed as the following:

F = MSG/MSE

If the null hypothesis is true and so all of the population means are equal, then this test statistic follows an F distribution I – 1 numerator degrees of freedom and N – I denominator degrees of freedom.

Example For the previous example, F = MSG/MSE = 52.87/7.62 ≈ 6.94.

The null hypothesis is rejected if the p-value is less than or equal to the level of significance α, or if F > F*I-1,N-I,α. The p-value is equal to P(Fdf₁,df₂ > F).

Example For the previous example, the p-value for the hypothesis test is p-value = P(F2,17 > 6.94) = (0.001,0.01), because 6.11 < F < 10.66.

Interpretations and Conclusion For ANOVA, the interpretation for the p-value and the conclusion are written differently.

For the p-value, the interpretation is: "If the mean (context of the mean) for all (I + context of the populations) is equal, then the probability of observing a value of the test statistic F at least as (high/low/extreme) as (F) would be (p-value)."

For the conclusion: "Since p-value (</>) α = (α), the null hypothesis is (rejected/failed to be rejected). At a 100(α)% level of significance, there is (sufficient/insufficient) evidence to conclude that at least one of the population means differs from the other means."

Example For the previous example, the p-value is interpreted: If the mean weight loss for all 3 diet programs is equal, then the probability of observing a value of the test statistic F at least as high as 6.94 would be between 0.001 and 0.01. The conclusion is then: Since p-value < α = 0.05, the null hypothesis is rejected. At a 5% level of significance, there is sufficient evidence to conclude that at least one of the population means differs from the other means.

Example For the previous example, the critical value method is used: The null hypothesis is rejected if F > F*2,17,0.05. Using the F* table, F* = 3.59. Since F = 6.94 > F*, the null hypothesis is rejected.

ANOVA Table The calculations for an analysis of variance F test can be summarized in an ANOVA table:

The total sum of squares, SSG + SSE, denoted as SST, measures the total variation in the data.

Equality of Population Standard Deviations Recall that it must be assumed that all I populations are normally distributed with a common standard deviation σ. Before a hypothesis test is conducted, a researcher should always examine the samples to assess whether these assumptions appear to be valid.

The researcher can get a good idea of what conclusion to expect from the hypothesis test by examining side-by-side box-plots for the I samples. The box-plots should have similar spreads, which helps verify the assumption of a common population standard deviation.

The assumption of normality is more difficult to verify with small samples, but it can be assumed normal if the box-plots are at least approximately symmetric with no outliers.

To check for the equality of population standard deviation, the same rule for a two-sample case is used:

max(sᵢ)/min(sᵢ)

where max(sᵢ) is the largest sample standard deviation out of all samples and min(sᵢ) is the smallest sample standard deviation out of all samples.

Just like with two-sample cases, if the quotient is less than or equal to two, then it can be assumed that the population standard deviations are equal and the F test can be used.

Steps of Conducting the F Test 1. State the level of significance. 2. State the hypotheses. H₀: μ₁ = μ₂ = ... = μᵢ vs. Hₐ: not all of the population means are equal 3. Determine whether the F test can be used. 4. Determine N, I, and the overall mean x̿. 5. Calculate SSG and SSE. 6. Calculate MSG and MSE. 7. Find the test statistic. 8. Construct an ANOVA table. 9. Write the decision rule. 10. Find the p-value, critical value, or confidence interval. 11. Interpret the p-value. (optional) 12. Write the conclusion.

Example Crown Plaza Hotel and Resorts offers special weekend rates at hotels and resorts across the U.S. Samples of properties from four regions of the country are selected, The room rate in dollars for each of the hotels is shown below. A researcher would like to test whether the true mean room rate differs for any of the four regions.

Then Population 1 is the Crown Plaza hotels in the Midwest, Population 2 is the Crown Plaza hotels in the Northeast, Population 3 is the Crown Plaza hotels in the South, and Population 4 is the Crown Plaza hotels in the West. Let the level of significance α = 0.05. H₀: μ₁ = μ₂ = μ₃ = μ₄ Hₐ: not all of the population means are equal The following are the side-by-side box-plots for each sample:

The distributions are all approximately symmetric, and so the assumption of normality appears to be reasonable. It also appears that the Northeast region may have a higher mean than the other regions. The hypothesis test will determine if the observed differences in means is significant. From the data, the following is calculated: n₁ = 5, x̄₁ = 141.80, s₁ = 38.25 n₂ = 7, x̄₂ = 192.14, s₂ = 34.67 n₃ = 9, x̄₃ = 166.22, s₃ = 41.63 n₄ = 6, x̄₄ = 153.67, s₄ = 23.08 Determining whether the F test can be used: max(sᵢ)/min(sᵢ) = 41.63/23.08 ≈ 1.80 < 2 Therefore, since it can be assumed the population standard deviations are equal, the F test can be used. Since there are I = 4 populations, the total sample size is N = 5 + 7 + 9 + 6 = 27. Calculating the overall mean: Σnᵢx̄ᵢ = (5)(141.80) + (7)(192.14) + (9)(166.22) + (6)(153.67) = 4471.98 x̿ = 4471.98/N =  4471.98/27 ≈ 165.63 Calculating the sum of squares for groups and error: SSG = Σnᵢ(x̄ᵢ - x̿)² = (5)(141.80 – 165.63)² + (7)(192.14 – 165.63)² + (9)(166.22 – 165.63)² + (6)(153.67 – 165.63)² ≈ 8620.19 SSE = Σ(nᵢ – 1)sᵢ² = (5 – 1)(38.25)² + (7 – 1)(34.67)² + (9 – 1)(41.63)² + (6 – 1)(23.08)² ≈ 29592.19 Calculating the mean square for groups and error: MSG = SSG/(I – 1) = 8620.19/3 ≈ 2873.40 MSE = SSE/(N – I) = 29592.19/23 ≈ 1286.62 Calculating the F test statistic: F = MSG/MSE = 2873.40/1286.62 ≈ 2.23 The following is the corresponding ANOVA table:

The null hypothesis is rejected if p-value ≤ α = 0.05. p-value = P(F3,23 > 2.23) > 0.10 (since F = 2.23 < 2.34, p-value is greater than 0.10) Since p-value > α = 0.05, the null hypothesis is failed to be rejected. At a 5% level of significance, there is insufficient evidence to conclude that at least one of the population means differs from the other means.

Example For the previous example, the critical value method is used: The null hypothesis is reject if F > F*3,23,0.05, where F* = 3.03. Since F = 2.23 < F*, the null hypothesis is failed to be rejected.

Example For the previous example, suppose the researcher conducted a two-sample test for the two means which are the furthest apart, which is Population 1 and Population 2. If the researcher concludes there is a difference between these two means, then he can say overall that at least one of the four means differs. However, when the researcher conducts the following test at a 5% level of significance: H₀: μ₁ = μ₂ Hₐ: μ₁ ≠ μ₂ The researcher gets a test statistic of t = 2.38 and a p-value of 0.0387. Then his conclusion is that the null hypothesis is rejected. However, this cannot overall conclude the four population means, because the previous example showed that at 5% level of significance, the population means were equal.

The above example shows why it is incorrect to look at the data and compare the two sample means that are the furthest apart from each other and conduct a two-sample t test. It is actually likely to see two of the sample means differ by a high amount. The more samples compared, the more likely some of the samples will have sample means further apart due to the principle of randomness.

Example A provincial education minister wants to determine whether there is any difference in the mean GPAs of students attending each of the province's three large universities. The GPAs of a sample of students from each of the universities are shown as the following:

Then Population 1 is the students in University A, Population 2 is the students in University B, and Population 3 is the students in University C. The following is its corresponding box-plots:

It appears there may be a difference between the mean GPAs for the students from the three universities. Additionally, since the distributions are approximately symmetric, it can be assumed normality is present. From the data, the following is calculated: n₁ = 5, x̄₁ = 2.91, s₁ = 0.344 n₂ = 6, x̄₂ = 3.74, s₂ = 0.471 n₃ = 4, x̄₃ = 3.31, s₃ = 0.596 Determining whether the F test can be used: max(sᵢ)/min(sᵢ) = 0.596/0.344 ≈ 1.73 < 2 Therefore, since it can be assumed the population standard deviations are equal, the F test can be used. Let the level of significance α = 0.10. H₀: μ₁ = μ₂ = μ₃ vs. Hₐ: not all of the population means are equal Since there are I = 3 populations, the total sample size is N = 5 + 6 + 4 = 15. Calculating the overall mean: Σnᵢx̄ᵢ = (5)(2.91) + (6)(3.74) + (4)(3.31) = 50.23 x̿ = 50.23/N = 50.23/15 ≈ 3.35 Calculating the sum of squares for groups and error: SSG = Σnᵢ(x̄ᵢ - x̿)² = (5)(2.91 – 3.35)² + (6)(3.74 – 3.35)² + (4)(3.31 – 3.35)² ≈ 1.89 SSE = Σ(nᵢ – 1)sᵢ² = (5 – 1)(0.344)² + (6 – 1)(0.471)² + (4 – 1)(0.596)² ≈ 2.65 Calculating the mean square for groups and error: MSG = SSG/(I – 1) = 1.89/2 ≈ 0.95 MSE = SSE/(N – I) = 2.65/12 ≈ 0.22 Calculating the F test statistic: F = MSG/MSE = 0.95/0.22 ≈ 4.32 The following is the corresponding ANOVA table:

The null hypothesis is rejected if p-value ≤ α = 0.10. p-value = P(F2,12 > 4.32) = (0.025,0.05) Since p-value < α = 0.10, the null hypothesis is rejected. At a 10% level of significance, there is sufficient evidence to conclude that at least one of the population mean GPAs differs from the other means.

Example For the previous example, the critical value method is used: The null hypothesis is reject if F > F*2,12,0.10, where F* = 2.81. Since F = 4.32 > F*, the null hypothesis is rejected.

Confidence Intervals and ANOVA To find a 100(1 – α)% confidence interval for one of the population means μᵢ, it is constructed as the following:

where sp² = MSE and t* is the upper α/2 critical value with N – I degrees of freedom.

Notice that this confidence interval uses a pooled estimate for the population standard deviation instead of the individual sample standard deviation. This is because one of the assumptions of ANOVA is that all populations have the same population standard deviation σ. To estimate this standard deviation, the pooled estimate is used, which is based on all of the samples rather than just the sample of interest.

Therefore, a better estimate of σ is obtained when all N observations are used to obtain sp rather than when just the nᵢ observations are used to obtain sᵢ.

MSE is the pooled estimate of the common variance σ² and √(MSE) is the pooled estimate of the common standard deviation σ.

Example For the previous example, a 95% confidence interval is constructed for University A's students' true mean GPA: x̄₁ = 2.91 Since α = 0.05, α/2 = 0.025. t* = 2.179 with 12 degrees of freedom. sp² = MSE = 0.22 (2.91 – (2.179)√(0.22/5),2.91 + (2.179)√(0.22/5)) ≈ (2.45,3.37)

Confidence Intervals for Two Samples of I Populations Confidence intervals for the difference between any two population means μᵢ – μj can be constructed as the following:

where t* has N – I degrees of freedom and sp² = MSE.

Example For the previous example, a 90% confidence interval is constructed for the difference in mean GPA for University B and University C: x̄₂ - x̄₃ = 3.74 – 3.31 = 0.43 Since α = 0.10, α/2 = 0.05. t* = 1.782 with 12 degrees of freedom. sp² = MSE = 0.22 sp²/n₂ + sp²/n₃ = 0.22/6 + 0.22/4 ≈ 0.092 √(0.092) ≈ 0.30 (0.43 – (1.782)(0.30),0.43 + (1.782)(0.30)) ≈ (-0.10,0.96)

Analysis of Variance vs. Pooled Two-Sample t Test Since ANOVA can be used for comparing two or more population means, ANOVA can be used as an alternative method for two-sample t test when only two populations need comparing and the population standard deviations are equal.

The following are three relationships between ANOVA's variables and pooled two-sample t test variables: 1. MSE = sp² 2. F = t² 3. p-value for ANOVA = p-value for pooled two-sample t test

Robustness of ANOVA Procedures Like the t procedures, the ANOVA procedures are also robust against non-normality.

The science of statistics deals with the collection, analysis, interpretation, and presentation of data.

Organizing and summarizing data is called descriptive statistics. Two ways to summarize data are by graphical methods and by using numbers and symbols. With knowledge on probability and probability distributions, formal methods, or inferential statistics, will be used to make conclusions from meaningful data. Statistical inference uses probability to determine how confident a researcher can be that their conclusions are correct.

Probability Probability is a mathematical tool used to study randomness. Probability deals with the chance, or likelihood, of an event occurring.

For example, if a fair coin was tossed, the outcomes may not be two heads and two tails. However, if the coin was tossed 4000 times, the outcomes will be close to 2000 heads and 2000 tails.

The expected theoretical probability of getting heads in one coin toss is 1/2 or 0.50. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions.

For example, the English statistician Karl Pearson tossed a coin 24,000 times with a result of 12,012 heads, which equals a probability of getting heads after 24,000 trials 0.5005.

Population and Sample In statistics, populations are generally studied. A population is a collection of persons, things, or objects under study. To study the population, a sample of the population is selected. The idea of sampling is to select a portion, or sample, to gain information about the population. Data are the result of sampling from a population. Because it is often difficult to examine an entire population, sampling is a practical technique.

Statistics and Parameters From the sample data, a statistic can be calculated. A statistic is a number that represents a measure of the sample. For example, the average number of points earned by the students in one math class at the end of the term is a statistic. The statistic is an estimate of a population parameter. A parameter is a number that represents a measure of the population. If all math classes in a school was the population, then the average number of points earned by the students in every math class at the end of the term is a parameter.

Representative Sample One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. This accuracy depends on how well the sample represents the population. The sample must contain the characteristics of the population to be a representative sample.

Variables A variable, denoted by capital letters such as X or Y, is a characteristic of interest for each unit in a population. Variables may be numerical or categorical. Numerical variables take on values that have equal units, such as weight in pounds or time in hours. Categorical variables place the object into a category.

For example, if X equals the number of points earned by one math student at the end of the term, then X is a numerical variable. If X is equal to a person's party affiliation, then some examples of X include Republican, Democrat, and Independent, and so X is then a categorical variable.

Mathematical calculations can be done to numerical variables, however mathematical calculations cannot be done to categorical variables.

Data Data are the actual values of the variable. The data can be in numbers or in words, where datum means a single value.

Mean and Proportion The words "mean" and "average" are often used interchangeably. If three exams in a math class were taken by a student, with each exam having the scores 86, 75, and 92, the mean score is calculated by adding the three exam scores together and dividing this sum by the total number of exams.

Any percentile can be converted into its corresponding z score by using the z score table. However, since this conversion from percentile to z score is important when doing z score and T score problems, a percentile table has been made.

Next to each percentile is its corresponding z score. Notice that all percentiles in the first two columns have a negative z score. This reinforces the fact that z scores that fall below the mean, or the 50th percentile, are negative. In the last two percentile columns, from the 51st percentile and up, are all positive z scores.

Finding Raw Data Values Using Percentiles and the z Score Equation When a percentile is known, the corresponding raw data value can also be determined by finding the corresponding z score to the given percentile and then using the equation x = zSD + M.

Although percentiles are easy to understand, they can lead to some distortions. This is because percentiles tend to compress the tails of the distribution, making extreme data values closer together than they actually are.

For example, a male adult height distribution has a mean of 69 inches and a standard deviation of 3.85. Comparing two heights, one at the 50th percentile and the other at the 54th percentile, a difference of 4 percentage points, the two corresponding z scores are found to be 0 and 0.10, respectively. Then the corresponding data values are calculated.

For the 50th percentile:

x = zSD + M = (0)(3.85) + 69 = 69

For the 54th percentile:

x = zSD + M = (0.10)(3.85) + 69 = 69.39

The difference between the two data values is 0.39 inches. Therefore, between the 50th and 54th percentile (percentiles in the center of the distribution), the difference in height is only 0.39 inches.

Now two new data values are compared. One at the 95th percentile and the other at the 99th percentile, which also differ by 4 percentage points, however both percentiles are at the end of the distribution (in this case, the high end) instead of at the center of the distribution.

The z scores for the 95th percentile and the 99th percentile is 1.65 and 2.41, respectively. The data values for each percentile is then calculated.

For the 95th percentile:

x = zSD + M = (1.65)(3.85) + 69 = 75.35

For the 99th percentile:

x = zSD + M = (2.41)(3.85) + 69 = 78.28

The difference between the two percentiles is approximately 3 inches, unlike the difference of 0.39 inches, despite both percentile pairs having the same difference of percentage points of 4.

Therefore, the difference between two percentiles in the center of the distribution is smaller than the difference between two percentiles at any end of the distribution.

Finding the Standard Deviation Using the z Score Equation The z score equation can also be rewritten to find the standard deviation.

Therefore, if a raw data value, the mean, and the corresponding z score are known, then the standard deviation can be determined. The standard deviation can also be determined if a percentile is given instead of the z score.

Note that, although the z score and the raw data value can be negative, the standard deviation can never be negative.

The Range Since the standard deviation can be found using the z score equation, the range can be approximated, because for normal distributions, the range is approximately six times the standard deviation.

Additionally, with the mean and the range known, the lowest value and the highest value of the distribution can be determined. Since the distribution is normal, one-half of the range falls above and below the mean.

To find the lowest value of the distribution, divide the range by two and subtract this quotient from the mean.

To find the highest value of the distribution, divide the range by two and add this quotient to the mean.

Finding the Mean Using the z Score Equation The z score equation can also be rewritten so that the mean can be determined if the standard deviation, a raw data value, and the corresponding z score (or percentile) are known.

Since the mean, the median, and the mode all fall at the same point under the normal curve, the median and the mode are equal to the mean.