Chapters
1-3
1-3
For each scenario identify the variable of measurement used:
A test is being conducted to assess the effectiveness of a new treatment that is intended to reduce chronic stress in individuals suffering from panic disorders: A randomly selected person is trained in the method and measured. What is the null hypothesis?
Those trained in the stress reduction method do not have lower stress scores.
How and why does using a one-tailed versus a two-tailed test affect power?
Using a one-tailed test instead of a two-tailed test increases statistical power because you're only testing in one direction, so the critical region is all on one side. This makes it easier to find a significant result if the effect is in the expected direction. But it also means you could miss an effect in the opposite direction, and it only makes sense to use a one-tailed test if you're sure the effect can't go the other way.
Name two assumptions you must check before running an ANOVA.
1) Homogeneity of variance (equal spread across groups) and 2) normal distribution of scores within each group.
Scenario: You are a researcher interested in examining whether the mean height of a sample of 100 students from a local high school differs significantly from the established population mean height of 65 inches. You collected height data from the sample and now want to determine if there is a statistically significant difference between the sample mean height and the population mean height.
A survey was conducted among 50 students to find out their favorite fruit. The following data shows test grades: 5, 7, 4, 5, 6, 5, & 4.
Construct a frequency table for the given data.
Value Frequency Percent
------ ---------- ------------
4 2 28.6
5 3 42.9
6 1 14.3
7 1 14.3
A pharmaceutical company is testing a new drug designed to lower blood pressure. They set up a clinical trial where they administer the drug to patients with hypertension. During the trial, the researchers conclude that there is no significant difference in blood pressure between the group taking the drug and the control group (placebo). However, in reality, the drug does effectively lower blood pressure. What type of error is this conclusion prone to?
Type II error
On a standard test, the population is known to have a mean of 500 and a standard deviation of 100. Those receiving an experimental treatment have a mean of 540. What is the effect size?
Effect Size = (Population1 M - Population 2 M) / Population SD = (540-500)/100 = .40
Why do you need to assume the populations have the same variance?
You need to assume the two groups are about equally spread out (same variance) because the t-test mixes their information together to figure out if the means are really different.
If one group’s scores are super spread out and the other group’s scores are really close together, it’s like comparing apples to oranges — the math won't work right, and you might get the wrong answer.
So, assuming they’re spread out about the same makes the test fair and the results trustworthy.
Scenario: As part of an educational study, you have two groups of students: Group A, consisting of 50 students who received tutoring sessions, and Group B, consisting of 50 students who did not receive any additional tutoring. You want to investigate whether there is a significant difference in the exam scores between these two groups.
Turn this z-score into a raw score:
In a standardized IQ test, the mean IQ score is 100, and the standard deviation is 15. Sarah's IQ score is 125.
Raw Score = 1.67
Do you reject the null hypothesis if the sample Score is –1.5 and the Cut-off score is –2.33?
What are degrees of freedom? How do you figure the degrees of freedom in a t-test for a single sample? What do they have to do with estimating the population variance? What do they have to do with the t distribution?
(a) The degrees of freedom are the number of scores free to vary.
(b)
The degrees of freedom in a t test for a single sample are the number of scores in the sample minus 1.
(c) In estimating the population variance, the formula is the sum of squared deviations divided by the degrees of freedom.
(d) t distributions differ slightly from each other according to the degrees of freedom.
List how hypothesis testing for a t-test for independent means is different from a t-test for dependent means in terms of Step 2, Step 3, and Step 4.
Step 2: Set the characteristics of the comparison distribution
Independent means: You calculate a pooled variance (combining the two groups' variances) and then find the standard error for the difference between the two means.
Dependent means: You focus on the difference scores (subtract one score from the other for each participant) and find the variance based on those difference scores.
Step 3: Determine the sample’s score on the comparison distribution
Independent means: You find the difference between the two sample means, then divide by the standard error.
Dependent means: You find the mean of the difference scores, then divide by the standard error of the difference scores.
Step 4: Make a decision
Independent means: Compare your t-score to the critical t based on degrees of freedom from both groups (n₁ + n₂ – 2).
Dependent means: Compare your t-score to the critical t using degrees of freedom based only on the number of pairs (n – 1).
Scenario: In a classroom intervention study, you administered a pre-test to 30 students to assess their math skills. Then, you implemented a new teaching method for several weeks and administered a post-test to the same group of students. Now, you want to determine if there is a significant difference in the mean scores between the pre-test and post-test.
If we find a correlation coefficient of 0.8 between the number of hours spent studying and the exam scores what does this mean?
It means that as the number of study hours increases, the exam scores tend to increase as well. This positive correlation suggests that studying more is associated with higher exam scores.
What are the steps of hypothesis testing?
What about the research situation makes the difference in whether you should carry out a z-test or a t-test for a single sample?
You use a z-test when the population mean and population standard deviation are both known. It’s usually for large, well-known populations where these numbers are already established.
You use a t-test when you don't know the population standard deviation and have to estimate it using your sample data. The t-test also adjusts for small sample sizes because smaller groups are more likely to vary by chance.
What is the purpose of conducting an independent groups ANOVA instead of multiple independent t-tests when comparing means of three or more groups?
Independent groups ANOVA is used to determine whether there are any statistically significant differences between the means of three or more independent groups simultaneously, while controlling for Type I error. Conducting multiple t-tests increases the chance of making a Type I error (false positive) due to the increased number of comparisons, whereas ANOVA controls for this by considering the overall variability between and within groups.
Scenario: You conducted a weight loss study where you randomly assigned 60 participants into two groups: Group A followed Diet Plan X, and Group B followed Diet Plan Y. After eight weeks, you recorded the weight loss for each participant. Now, you want to determine if there is a significant difference in the average weight loss between the two diet plans.
Find the variance for the following set of numbers: 5, 8, 10, 12.
A researcher predicts that showing a certain video will change people's attitudes toward alcohol. The researcher then randomly selects 36 people, shows them the video, and gives them an attitude questionnaire. The mean score on the attitude test for the 36 people is 70, while the mean score for people in general on the test is 75, with a standard deviation of 12. Using the five steps of hypothesis testing and a 5% significance level, carry out a Z-test to determine if watching the video changes people's attitudes toward alcohol.
Step 1: State the Hypotheses
H₀: The mean attitude of people shown the video is the same as the general population.
H₁: The mean attitude of people shown the video is different from the general population.
Step 2: Characteristics of the Comparison Distribution
Population Mean (µ) = 75
Standard Deviation = {12^2 / 36}=2
Step 3: Cutoff score
Cutoff for 5% significance level (two-tailed): ±1.96
Step 4: Sample Z-score
z={70 - 75}/{2} = -2.50
Step 5: Make a Decision
The z-score (-2.50) is more extreme than -1.96, so reject the null hypothesis. Seeing the video does change attitudes toward alcohol.
In a t-test for dependent means what is usually considered to be the mean of the “known” population and why?
The mean of the “known” population is 0. This is because you are comparing your sample to a situation in which there is no difference— a population of difference scores in which the average difference is 0
What does the F-statistic in ANOVA represent, and how is it calculated?
The F-statistic in ANOVA tests the ratio of the variance between the groups to the variance within the groups. It is calculated by dividing the between-group variance (MSB) by the within-group variance (MSW).
Scenario: As part of a fitness study, you recruited 90 participants and randomly assigned them to three different exercise intensity groups: Low, Medium, and High. After six weeks of following their assigned exercise regimen, you measured their heart rates. Now, you want to examine if there is a significant difference in the average heart rates among the three exercise intensity groups.