Please feel free to use any part of this and even freer to email me back with suggestions and misconceptions I have imposed on my students. This test went well but not as well as previous tests. I noticed that my students did not do as good a job intrepreting the results of tests as they have in the past. In addition, they did not pick up the need to interpet confidence intervals as providing information about magnitudes in addition to duality with tests of significance.
1. Circle the correct answer: A confidence interval is robust when:
a) Changes in the confidence level do not affect the confidence interval very much. b) The confidence interval does not change if the assumptions of the procedure are violated. c) The confidence interval does not change very much if the assumptions of the procedure are violated. d) The confidence interval is large.
2. Circle the correct answer: The data in a two-sided two-proportion test of significance produces a statistically significant result. Then a 95% confidence interval is computed.
a) About 95% of the data should be contained in the interval. b) About 95% of the time we expect the true population proportion to be in the interval. c) Zero is not in the interval. d) Zero is in the interval. e) None of the above.
3. Demonstrate your knowledge of the meaning of Standard Error in explaining whether standard error or standard deviation is used in doing inference for a mean of a population after gathering a sample from an unknown population.
4. Explain why you would be very concerned about a study which reported statistically significant results (p = 0.001, df = 0) after doing a one-sample t-test for the mean of a population. Be specific.
5. Explain why it is necessary to have the t-distributions?
6. A student searched the Internet for data for a project. He found a file with the population of each of the states of the USA. He used his calculator to construct a 95% confidence interval about the mean of the populations. Given the data, why does this not make sense?
7. Explain why the pooled proportion, p(c), is used in a two-proportion test of significance but is not used in calculating the corresponding confidence interval.
8. An experiment is performed in which the proportion of males in AP Statistics Classes is compared to the proportion of males in AP Calculus classes. A two-sided two-proportion test of significance is performed with Ho: Theta(Stats) = Theta(calc). The test produces a statistically significant result with a confidence interval of (-0.3, -0.1) where t = (p-hat(stats)-p-hat(calc))/root[(p-hat(c))(1- p-hat(c))(1/n1+1/n2)]
What additional information not already stated above is provided by the confidence interval? Your answer should be as specific as possible.
9. The weight of 9 men have mean 175 lbs. and standard deviation 15 lbs. Calculate the standard error of the mean.
10. It is logical to assume that a dirty air filter will reduce the gas mileage of a car. In an experiment 6 pairs of identical cars are rented. In one car of each pair a brand NEW air filter is installed. On the other car of each pair a DIRTY air filter (of the same type) is installed. Gas mileage for these cars during the experiment is listed below:
Pairs Gas Mileage with Clean Filter Gas Mileage with Dirty Filter Small Chevys 23 21 Medium Chevys 22 20 Large Chevys 19 17 Small Fords 25 26 Medium Fords 25 21 Large Fords 25 23
a) What is the name of the most appropriate test of significance to apply to see if dirty air filters reduce the gas mileage of a sensible population of cars?
b) While supplying just enough information to keep your teacher informed of what you are doing, use your calculator to perform the computations involved in the test you just mentioned. Include at some point the formula for the test statistic.
c) Carefully and fully report the results of the test in a meaningful way.
11. Cuckoos lay their eggs in the nests of other birds. Some biologists speculate that the size of the cuckoo's eggs might be different depending on whether the eggs are laid in warbler's nests or wren's nests. To check this, biologist searched a wildlife refuge for warblers' and wrens' nests. Summary statistics for the lengths (in mm) of cuckoo's eggs found in the nest of warblers and wrens are shown below:
From Warblers' nests: n1 = 29, x-bar=21.62, s1= 0.65 From Wren's nests: n2 =35, x-bar = 21.12, s2=0.75
a. Demonstrate you can do the following BY HAND and with tables:
Do a fully annotated test of significance to determine if the biologists' speculation that the size of the eggs differ depending on whether they are laid in warblers' nest or wrens' nests.
b. Demonstrate you can do the following BY HAND and with tables:
Determine and fully interpret a 96% confidence interval for the above problem.