and that he was going to win the election. Statistically speaking (say at the 0.05 level), how happy
should Dopey be (i.e., how sure is he that support for Grumpy has dropped)?
Consider, once again, the situation of question #7 above. In that problem, a one-sided, two-
proportion z -test was conducted to determine if there had been a drop in the proportion of people
who watch the show “I Want to Marry a Statistician” when it was moved from Monday to Wednesday
evenings.
Suppose instead that the producers were interested in whether the popularity ratings for the show had
changed in any direction since the move. Over the seasons the show had been broadcast on Mondays,
the popularity rating for the show (10 high, 1 low) had averaged 7.3. After moving the show to the
new time, ratings were taken for 12 consecutive weeks. The average rating was determined to be 6.1
with a sample standard deviation of 2.7. Does this provide evidence, at the 0.05 level of significance,
that the ratings for the show have changed? Use a confidence interval, rather than a t -test, as part of
your argument. A dotplot of the data indicates that the ratings are approximately normally distributed.
- A two-sample study for the difference between two population means will utilize t -procedures and
is to be done at the 0.05 level of significance. The sample sizes are 23 and 27. What is the upper
critical value (t *) for the rejection region if
(a) the alternative hypothesis is one-sided, and the conservative method is used to determine the
degrees of freedom?
(b) the alternative hypothesis is two-sided and the conservative method is used to determine the
degrees of freedom?
(c) the alternative hypothesis is one-sided and you assume that the population variances are equal?
(d) the alternative hypothesis is two-sided, and you assume that the population variances are equal?
Cumulative Review Problems
How large a sample is needed to estimate a population proportion within 2.5% at the 99% level of
confidence if
a. you have no reasonable estimate of the population proportion?
b. you have data that show the population proportion should be about 0.7?
- Let X be a binomial random variable with n = 250 and p = 0.6. Use a normal approximation to the
binomial to approximate P (X > 160). - Write the mathematical expression you would use to evaluate P (X > 2) for a binomial random
variable X that has B (5, 0.3) (that is, X is a binomial random variable equal to the number of
successes out of 5 trials of an event that occurs with probability of success p = 0.3). Do not evaluate. - An individual is picked at random from a group of 55 office workers. Thirty of the workers are
female, and 25 are male. Six of the women are administrators. Given that the individual picked is
female, what is the probability she is an administrator? - A random sample of 25 cigarettes of a certain brand were tested for nicotine content, and the mean
was found to be 1.85 mg with a standard deviation of 0.75 mg. Find a 90% confidence interval for the
mean number of mg of nicotine in this type of cigarette. Assume that the amount of nicotine in
cigarettes is approximately normally distributed. Interpret the interval in the context of the problem.