problem: 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.
problem: 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, find out a 95% confidence interval for the true proportion of lake trout with lamprey scars.
problem: 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.
problem: 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) find out the p-value for the test statistic value in (a) and interpret it.
problem: 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.