For this lesson, we will be comparing two group population values
tutor4helpyouPractice Problems: Chapter 10
For this lesson, we will be comparing two group population values (or two paired observations for those in the population). There are some subtle differences in the formulas used for computing the Standard Error used in both our Confidence Intervals of the Difference and in our Hypothesis Tests which are noted in the instructions below.
1) We wish to compare the population proportion of those graduates who pass the Bar Exam on the first try for two small law schools. The researcher has randomly selected graduates from the previous 4 years graduating classes and asked those selected if they passed the Bar Exam on the first try. Law School A had 150 out of 200 sampled who indicated that they passed the Bar Exam on the first try. Law School B had 170 out of 250 sampled who indicated that they passed the Bar Exam on the first try. We wish to see if there is a difference between the population proportions of those who passed the Bar Exam on the first try for the two law schools (p1 is for School A, p2 is for School B).
We use this SE formula for confidence intervals of the difference between population proportions.
i) Calculate your SE. Show all work
ii) What is your Z multiplier?
iii) Construct your 95% Confidence Interval of the Difference. Show all work and indicate the endpoints.
iv) Is there a difference in the two group population proportions? Why or why not?
B)Now, do a Hypothesis Test to see if there is a difference between the two population proportions.
i) Since we are testing to see if there is a difference, our alternative is that there is a difference and our null is that there is no difference:
ii) Calculate our Z test statistic. Show all work.
iii) What is the p-value of our Z test statistic? Show all work.
iv) What is your conclusion and why?
A) Construct a 95% Confidence Interval of the Difference between the two group population means.
i) Calculate your SE. Show all work
ii) What is your t multiplier from the t Table?
iii) Construct your 95% Confidence Interval of the Difference. Show all work and indicate the endpoints.
iv) Is there a difference in the two group population means? Why or why not?
B) Now, do a Hypothesis Test to see if there is a difference between the two population means.
i) What are the null and alternative hypotheses? Use proper statistical notation (use proper symbols for population means)
ii) We use the same SE as we used for the Confidence Interval of the Difference in part A. Calculate your t test statistic. Show all work.
iii) Using the t table in the text. Indicate the range of the p-value for your t test statistic.
iv) What is your conclusion and why?
v) Part iv is for a conclusion regarding statistical significance. Is the difference between the groups of practical significance? Assume a total possible score of 60. Look at your CI of the Difference in part A to help answer this question.
3) A certain diet has been effective in past studies. We want to see if a certain diet is effective for our group. The duration of the study is 8 weeks. Study participants had their beginning weight measured and then were measured again 8 weeks after starting the diet. The change in weight (beginning weight - ending weight) for each participant is the response variable. To simplify our calculations, we will do our analysis on only 4 pairs of observations. Note the difference between this and the scenario in Question 2. In Question 2 we were comparing a change in scores between two independent groups and the Hypothesis Test we used was a 2-sample t-test. In this problem we are comparing the change per person and not comparing between groups. The Hypothesis Test we use is a Paired t-test.
Below are the observations taken on the four participants.
participant |
beginning weight |
ending weight |
difference (beg wt-end wt) |
1 |
210 |
200 |
10 |
2 |
150 |
145 |
5 |
3 |
185 |
160 |
25 |
4 |
180 |
184 |
-4 |
A) We test to see if the diet was effective. If the diet was effective, then the difference between the beginning and ending weight would be positive since the ending weight would be less than the beginning weight.
i) What are the null and alternative hypotheses? Use proper statistical notation for paired quantitative data (μd.)
ii)
Show all work.
iii)
iv) Calculate your t test statistic. Show all work.
v) Using the t table in the text calculate your p-value range. Remember,
DF = n-1 where n is the number of pairs.
vi) What is your conclusion and why?
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