Statistical Based Inference

A representative of a car manufacturer in the United States made the following claim in a news report. "Ten years ago, only 53 percent of Americans owned American-made cars, but that figure is significantly higher today." A research group conducted a study to investigate whether the claim was true. The group found that 56 percent of a randomly selected sample of car owners in the United States owned American-made cars. A test of the appropriate hypotheses resulted in a p-value of 0.283. Assuming the conditions for inference were met, is there sufficient evidence to conclude, at the significance level of a = 0.05, that the proportion of all car owners in the United States who own American-made cars has increased from what it was ten years ago?

• Yes, because 0.56 > 0.53.

• Yes, because a reasonable interval for the proportion is 0.56 ± 0.283.

• Yes, because 0.56 - 0.53 = 0.03 and 0.03 < 0.05.

• No, because 0.283 < 0.53.

• No, because 0.283 > 0.05.

A botanist collected one leaf at random from each of 10 randomly selected mature maple trees of the same species. The mean and the standard deviation of the surface areas for the 10 leaves in the sample were computed.Assume the distribution of surface areas of maple leaves is normal. What is the appropriate method for constructing a one-sample confidence interval to estimate the population mean surface area of the species of maple leaves, and why is the method appropriate?

• The t-interval is appropriate, because the population standard deviation is not known.

• The t-interval is appropriate, because the t-interval is narrower than the z-interval.

• The z-interval is appropriate, because the z-interval is narrower than the t-interval.

• The z-interval is appropriate, because the central limit theorem applies.

• The z-interval is appropriate, because the sample standard deviation is known.

A commercial for a breakfast cereal is shown during a certain television program. The manufacturer of the cereal wants to estimate the percent of television viewers who watch the program. The manufacturer wants the estimate to have a margin of error of at most 0.02 at a level of 95 percent confidence. Of the following, which is the smallest sample size that will satisfy the manufacturer’s requirements?

• 40

• 50

• 100

• 1700

• 2500

Meteorologists are interested in the relationship between minimum pressure and maximum wind speed of hurricanes. The minimum pressure, in millibars, and maximum wind speed, in knots, were collected for a random sample of 100 hurricanes from the year 1995 to the year 2012. A regression analysis of maximum wind speed on minimum wind pressure produced a 95 percent confidence interval of (-1.42, -1.20) for the slope of the least-squares regression line. Which statement is a correct interpretation of the interval?

• The probability is 0.95 that wind speed will decrease, on average, between 1.20 knots and 1.42 knots for each millibar increase in minimum pressure.

• The probability is 0.95 that a different sample of 100 hurricanes will result in an increase, on average, of wind speed between 1.20 knots and 1.42 knots for each millibar increase in minimum pressure.

• We can be 95% confident that wind speed decreases, on average, between 1.20 knots and 1.42 knots for each millibar increase in minimum pressure.

• We can be 95% confident that wind speed increases, on average, between 1.20 knots and 1.42 knots for each millibar increase in minimum pressure.

• We can be 95% confident that, for any sample of hurricanes, the wind speed will decrease, on average, between 1.20 knots and 1.42 knots for each millibar increase in minimum pressure.

On the day before an election in a large city, each person in a random sample of 1,000 likely voters is asked which candidate he or she plans to vote for. Of the people in the sample, 55 percent say they will vote for candidate Taylor. A margin of error of 3 percentage points is calculated. Which of the following statements is appropriate?

• The proportion of all likely voters who plan to vote for candidate Taylor must be the same as the proportion of voters in the sample who plan to vote for candidate Taylor (55 percent), because the data were collected from a random sample.

• The sample proportion minus the margin of error is greater than 0.50, which provides evidence that more than half of all likely voters plan to vote for candidate Taylor.

• It is not possible to draw any conclusion about the proportion of all likely voters who plan to vote for candidate Taylor because the 1,000 likely voters in the sample represent only a small fraction of all likely voters in a large city.

• It is not possible to draw any conclusion about the proportion of all likely voters who plan to vote for candidate Taylor because this is not an experiment.

• It is not possible to draw any conclusion about the proportion of all likely voters who plan to vote for candidate Taylor because this is a random sample and not a census.

A 90 percent confidence interval for the slope of a regression line is determined to be (-0.181, 1.529). Which of the following statements must be true?

• The correlation coefficient of the data is positive.

• The sum of the residuals for the data based on the regression line is positive.

• A scatterplot of the data would show a linear pattern.

• The slope of the sample regression line is 1.348.

• The slope of the sample regression line is 0.

An airline claims that the mean flight time between City X and City Y is 38 minutes. After taking many flights, a local business group believes that the claim is unrealistic and that the actual mean flight time is greater than 38 minutes. If the group conducts a study to investigate its belief, which of the following hypotheses should be tested?

• 
• 
• 
• 
• 

A national health study reported that the proportion of students with elevated blood pressure is 0.15. The principal of a local high school believes that the proportion of students in the school with elevated blood pressure is greater than 0.15. If a large random sample is used, which of the following is the most appropriate test to investigate the principal’s belief?

• z-test for a proportion

• z-test for a difference between two proportions

• chi-square test for homogeneity of proportions

• t-test for a mean

• matched-pairs t-test

A school administrator is interested in estimating the proportion of students in the district who participate in community service activities. From a random sample of 100 students in the district, the administrator will construct a 99 percent confidence interval for the proportion of all district students who participate in community service activities. Which of the following statements must be true?

• The population proportion will be in the confidence interval.

• The probability that the confidence interval will include the population proportion is 0.99.

• The probability that the confidence interval will include the sample proportion is 0.99.

• The population proportion and the sample proportion will be equal

• The probability that the population proportion and the sample proportion will be equal is 0.99.

A large city newspaper periodically reports the mean cost of dinner for two people at restaurants in the city. The newspaper staff will collect data from a random sample of restaurants in the city and estimate the mean price using a 90 percent confidence interval. In past years, the standard deviation has always been very close to $35. Assuming that the population standard deviation is $35, which of the following is the minimum sample size needed to obtain a margin of error of no more than $5 ?

• 90

• 112

• 133

• 147

• 195