Question: Which of the following statements are correct interpretations of a 95% confidence interval for μ?
(a) 95% of the observations in the sample will be contained in the confidence interval.
(b) 95% of the population will be contained in the confidence interval.
(c) 95% of the sample means will be contained in the confidence interval.
(d) Before the random sample was taken, there was a 95% probability of getting a sample which would give a confidence interval which contained μ. Thus, we are 95% confident our intervalcontains μ.
(e) 95% of the population means will be contained in the confidence interval.
(f) If repeated samples were taken, we would expect that 95% of the confidence intervals would contain μ and 5% would not.
Question: In a certain lake a limnologist wishes to estimate the proportion of lake trout with lamprey scars.
(a) How large of a random sample should be taken if the limnologist has no prior knowledge of the true proportion, but wants the bound to be within 0.04 with 90% confidence?
(b) How large should the sample be if the limnologist knows the true proportion does not exceed 0.20?
(c) If the limnologist caught 326 lake trout, and 47 had scars, calculate a 95% confidence interval for the true proportion of lake trout with lamprey scars.
Question: Obtain the critical value (or values) for a hypothesis test for the mean if you conduct:
(a) an upper tail “z” test at the 1% level of significance.
(b) a two-tailed “z” test at the 1% level of significance.
(c) a lower tail “z” test at the 5% level of significance.
(d) a two-tailed “z” test at the 5% level of significance.
(e) an upper tail “z” test at the 10% level of significance.
(f) a upper tail “t” test with 17 d.f. at the 5% level of significance.
(g) a two-tailed “t” test with 17 d.f. at the 5% level of significance.
(h) a lower tail “t” test with 6 d.f. at the 1% level of significance.
(i) a two-tailed “t” test with 6 d.f. at the 10% level of significance.
(j) an upper tail “t” test with 6 d.f. at the 5% level of significance.
Question: A subject is asked to pick the suit of a card that she does not see, picked at random from a standard deck of playing cards. If the subject has no extra-sensory powers (ESO) her guesses should be correct, on average, 1 in 4 times. However, if she has ESP then the proportion of correct guesses should be better than 1 in 4.
(a) If the subject gets the suit correct 120 times in 400 trials, test (at the 5% level of significance) the statistical evidence that the subject might have ESP.
(b) Calculate the p-value for the test statistic value in (a) and interpret it.
Question: A study looked at the relationship between coronary heart disease (CHD) and coffee consumption in a group of 40–55 year old men. Among the 790 heavy coffee drinkers (at least 100 cups per month), there were 38 CHD cases. Among the 928 moderate and non-drinkers (less than 100 cups per month), there were 39 CHD cases.
(a) Test the hypothesis that the rate of CHD is higher for heavy coffee drinkers at the 5% level of significance.
(b) Obtain the p-value for the value of the test statistic in (b).
(c) What would the p-value be if the hypothesis to be tested was simply that the rate of CHD differs between the two groups?
(d) Obtain a 95% confidence interval for the difference in the population proportions of CHD cases between the two categories of coffee drinkers.