(4 points) Consider the following frequency distribution table:

X f
16 1
5 2
4 3
3 3
2 5
1 6

Find the sample size, mean, median, and mode for the set of scores.





Among the sample mean, median, and mode for the example data, which best reflects the “center” of the data?  Justify your answer.





If the score of X=16 was removed from the set of scores, how would the mean be affected?





Use software to create a histogram for the set of scores.  (Please attach the histogram to your exam.)

Probability
(4 points) Consider the following standard normal distribution:

Without looking up exact probabilities, how can you tell that p(z<-0.5)<0.5?  That is, how do we know the shaded area is less than 0.5?





Using a table or software, find the exact probabilities for the two sections of the normal distribution.  That is, find:
p(z<-0.5)
p(z>-0.5)





What does the Central Limit Theorem tell us about the sampling distribution of sample means, when N≥30?





Suppose that μ=10, and that the standard error of the sample mean is 3.  Give an interpretation of the sample mean, using these values.

Inferential Statistics
(6 points) Testing with one sample mean. A research team is studying the effects of exercise on blood pressure. The team recruits a sample of N=408 study participants from a local community, and wants to see if the local adults are different than typical adults in any way. An example of a way the sample might be different is in physical characteristics, such as height. According to the Centers for Disease Control, the mean height of US adults aged 20 and older is about 66.5 inches. For the sample of local adults, the mean is M=67.0 inches, and the sample standard deviation is s=5.3 inches.

Use a hypothesis test with α=.05 to test whether local adults are taller or shorter than all US adults.

Is a z-test or a t-test more appropriate here and why?




A two-tailed test is more appropriate than a one-tailed test here.  Why?




Write the null and alternative hypotheses.  Assume this is a two-tailed test.




Calculate the test statistic that corresponds to M=67.0 inches, and the corresponding p-value.




Sketch a distribution for the test statistic under the null hypothesis, mark the test statistic value, and shade the region that corresponds to the p-value.  (Please attach the sketch to your test.)




Make a decision about H_0 and state the conclusion.





(6 points) Testing with two sample means.  In the article “Teaching well-being at scale: An intervention study” (Yaden et al., 2021), the researchers study the difference in subjective well-being for students upon completion of a treatment class (The Science of Well-Being) or a control class (Introduction to Psychology).  The outcome variable was the “overall well-being” variable of the PERMA Profiler, on a 0-10 scale.  In the table below are some descriptive statistics for the variable (pp. 4-5).  (The sample sizes are very large because these classes were offered for free online.)

Table 1.
Group Sample Size Mean Standard Deviation
Treatment (Group 1) N_1=1,228 7.73 1.17
Control (Group 2) N_2=1,480 7.27 1.30

The researchers found the difference in sample means to be statistically significant (p<0.001), with an effect size of d=0.37.

The following questions are about differences in population means for the treatment and control groups.

Is an independent-samples test or a paired-samples test more appropriate here and why? 




Write the null and alternative hypotheses.  Assume this is a two-tailed test.




Calculate the standard error and the test statistic.




Find the p-value and interpret it.  Confirm that it is less than 0.001.



Calculate the pooled standard deviation (use Google Sheets), and use it to compute Cohen’s d.  Your value should be very close to the value of d=0.37 found by the researchers, but may be slightly different due to rounding.



Will a 95% confidence interval for μ_treatment-μ_control contain 0?  Justify your answer.  (Hint: you were asked to calculate such a confidence interval for Midterm #3.)




(2 points) Choosing an appropriate statistical test. For the following scenarios, choose which statistical test would be most appropriate (pick A, B, C, or D): 
A. One-sample z-test (one sample, one measurement) 
B. One-sample t-test (one sample, one measurement)
C. Independent samples t-test (two samples, one measurement each) 
D. Paired samples t-test (one sample, two measurements)

A famous recipe for chocolate chip cookies includes a small amount of lemon juice.  A famous cook, Martha, is skeptical the lemon juice makes the cookies taste better.  So, she bakes two batches of the cookies: one with the lemon-juice, and one without.  Then, she recruits two different sets of volunteers: one set gets the lemon-juice cookies, and one set gets the cookies without lemon-juice.  Martha then compares the ratings for the two cookies.

Choice for a) _

A high school chemistry teacher wants her students to memorize the elements of the periodic table.  She decides that having students listen to “The Periodic Table Song” in her class every day for a week will help.  Before the “treatment” of listening, she gives the students a quiz on the periodic table.  Then, after the “treatment” of listening for a week, she gives the students the same quiz.

Choice for b) _