Chapter 8: Estimating with Confidence

The U.S. Forest Service is considering additional restrictions on the number of vehicles allowed to enter Yellowstone National Park. To assess public reaction, the service asks a random sample of 150 visitors if they favor the proposal, Of these, 89 say "Yes."

a. Construct and interpret a $99\%$confidence interval for the proportion of all visitors to Yellowstone who favor the restrictions.

b. Based on your work in part (a), is there convincing evidence that more than half of all visitors to Yellowstone National Park favor the proposal? Justify your answer.

A milk processor monitors the number of bacteria per milliliter in raw milk received at the factory. A random sample of $10$one-milliliter specimens of milk supplied by one producer gives the following data:

Construct and interpret a $90\%$ confidence interval for the population mean $\mu$.

The weights (in pounds) of three adult males are$160$,$215$and$195$. What is the standard error of the mean for these data?

a. $190$

b. $27.84$

c. $22.73$

d. $16.07$

e.$13.13$

In preparing to construct a one-sample t interval for a population mean, suppose we are not sure if the population distribution is Normal. In which of the following circumstances would we not be safe constructing the interval based on an SRS of size$14$from the population?

a. A stemplot of the data is roughly bell-shaped.

b. A histogram of the data shows slight skewness.

c. A boxplot shows that the values above the median are much more variable than the values below the median.

d. The sample standard deviation is large.

e. The sample standard deviation is small.

Many television viewers express doubts about the validity of certain commercials. In an attempt to answer their critics, Timex Group USA wishes to estimate the true proportion p of all consumers who believe what is shown in Timex television commercials. What is the smallest number of consumers that Timex can survey to guarantee a margin of error of $0.05$or less at a $99\%$confidence level?

a. $550$

b. $600$

c. $650$

d. $700$

e. $750$

You want to compute a $90\%$confidence interval for the mean of a population with an unknown population standard deviation. The sample size is $30$. What critical value should you use for this interval?

a. $1.645$

b. $1.699$

c. $1.697$

d. $1.96$

e. $2.045$

A radio talk show host with a large audience is interested in the proportion p of adults in his listening area who think the drinking age should be lowered to eighteen. To find this out, he poses the following question to his listeners: “Do you think that the drinking age should be reduced to eighteen in light of the fact that 18-year-olds are eligible for military service?” He asks listeners to go to his website and vote “Yes” if they agree the drinking age should be lowered and “No” if not. Of the 100 people who voted, 70 answered “Yes.” Which of the following conditions are violated?

I. Random

II. 10%

III. Large Counts

a. I only

b. II only

c. III only

d. I and II only

e. I, II, and III

A $90\%$confidence interval for the mean μ of a population is computed from a random sample and is found to be$90\pm 30$. Which of the following could be the $95\%$confidence interval based on the same data?

a. $90\pm 21$

b. $90\pm 30$

c. $90\pm 39$

d. $90\pm 70$

e. Without knowing the sample size, any of the above answers could be the $95\%$confidence interval.

Suppose we want a 90% confidence interval for the average amount spent on books by freshmen in their first year at a major university. The interval is to have a margin of error of at most $2.Basedonlastyea{r}^{\prime }sbooksales,weestimatethatthestandarddeviationoftheamountspentwillbecloseto$30. The number of observations required is closest to which of the following?

a. $25$

b.$30$

c. $225$

d. $609$

e. $865$

A telephone poll of an SRS of 1234 adults found that 62% are generally satisfied with their lives. The announced margin of error for the poll was 3%. Does the margin of error account for the fact that some adults do not have telephones?

a. Yes; the margin of error accounts for all sources of error in the poll.

b. Yes; taking an SRS eliminates any possible bias in estimating the population proportion.

c. Yes; the margin of error accounts for undercoverage but not nonresponse.

d. No; the margin of error accounts for nonresponse but not undercoverage.

e. No; the margin of error only accounts for sampling variability.