Section 1 – Statistical Inference and Confidence Intervals
Question 1 (10 Marks)
The owner of a sporting goods store surveys a group of customers on (i) the average amount spent, and (ii) whether they purchased at least one item of clothing.
The results of a sample of 80 customers are:
Average amount spent: X ¯=49.50
Standard deviation: S=9.50.
Number who bought an item of supporter’s apparel: 20.
Construct a 95% confidence interval for the mean amount spent per shopper.
Construct a 95% confidence interval for the proportion of shoppers who bought at least one item of supporter’s apparel.
The owner of a competing store decides to conduct his own survey.
What sample size does he need if he wants to estimate a 90% confidence interval for the mean amount spent to within ±$3.00, assuming a standard deviation of $8.50?
What sample size does he need to estimate a 90% confidence interval for the proportion who purchase at least one item of supporter’s apparel to within ±0.05?
Based on your answers to c) and d), how large a sample should the store owner use?
Section 2 – Hypothesis Testing: One-Sample Tests
Question 2 (5 Marks)
You work for a company that tests new pharmaceutical products. You are currently testing a new product. From a sample of 300 subjects, 184 report an improvement in their symptoms from taking the new product. The existing product improves the symptoms for 60% of people who take it.
State the null and alternative hypotheses that you would use to test the effectiveness of the new product.
Explain the risks associated with Type 1 and Type 2 errors in this context.
Which of the errors is more serious in this context? Give reasons for your answer.
Question 3 (5 Marks)
Back at the sports store from question 1, the owner decides to use significance tests to better understand the behaviour of his customers. Starting with a new sample of 60 customers, he finds:
Average amount spent: X ¯= 42.8
Standard deviation: S = 11.7
14 customers purchased an item of supporter’s apparel.
At the 5% significance level, is there evidence that the mean spend is different from $40.00? State the null and alternative hypothesis.
Find the p-value in a).
At the 5% level, is there evidence that fewer than 10% of customers purchase at least one item of supporter’s apparel.
Section 3 – Two Sample Tests
Question 4 (10 Marks)
A researcher claims that New Zealanders spend more time exercising than Australians. The researcher takes a random sample of 41 New Zealanders and finds that they spend an average of 130 minutes per day exercising, with a standard deviation of 24. A sample of 61 Australians averaged 110 minutes a day of exercise, with a standard deviation of 16.
Assuming equal variances, test the claim that New Zealanders exercise more than Australians at the 0.05 significance level.
Assuming unequal variances, repeat the test at the 0.05 significance level.
Test whether the variances are equal at the 0.05 significance level.
Based on your answer to part c), which was the appropriate test for the means (a), or b))? Give reasons for your answer.
Explain the unequal variances problem and its relevance, if any, to your conclusions about whether New Zealanders exercise more than Australians.
Section 4 – Analysis of Variance
Question 5 (6 Marks)
Test, at the 5% significance level, to discover whether differences exist between the population means given the statistics below. Remember to state the null and alternative hypotheses clearly.
Group 1 Group 2 Group 3
12 18 19
7 17 15
14 7 18
15 10 16
Question 6 (4 Marks)
A researcher studying four different populations proposes the following:
H_0 : µ_1=µ_2=µ_3=µ_4
H_1 : not all µ_i are equal (i=1,2,3,4)
He takes samples from each of the populations, obtaining the following data:
Group 1 Group 2 Group 3 Group 4
Sample mean 110 105 91 102
Sample size 8 9 5 6
The within-group variation of the data (SSW) is equal to 108.
Using the Tukey-Kramer procedure, calculate the critical range for making a comparison of groups 1 and 4.
Using the data and your answer to part (a), explain whether or not the null hypothesis should be rejected.
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