Mean Inference AP Stats

Given a confidence interval (0.05, 0.13), what is the margin of error?

• 0.05

• 0.13

• 0.09

• 0.04

A machine at a manufacturing company is programmed to fill shampoo bottles such that the amount of shampoo in each bottle is normally distributed with mean 0.60 liter and standard deviation 0.04 liter. Let the random variable A represent the amount of shampoo, in liters, that is inserted into a bottle by the filling machine. A bottle is considered underfilled if it has less than 0.50 liter of shampoo. Determine the probability that a randomly selected bottle of shampoo will be underfilled.

• .9938

• .0062

• -2.5

• none of these

The most important condition for making an inference about a population mean from a significance test is that

• the data comes from a radom sample

• the population distribution is exactly normal

• the data contains no outliers

• the sample size is less than 10% of the population

• the sample size is at least 30

Which of the following is 𝑛𝑜𝑡 one of the conditions that must be satisfied in order to perform inference about the slope of a least-squares regression line?

• The standard deviation 𝜎 of the population of 𝑦-values corresponding to a given value of 𝑥 is always the same, regardless of the specific value of 𝑥.

• The data come from a random sample or a randomized experiment.

• The sample size—that is, the number of paired observations (𝑥,𝑦)—exceeds 30.

• For each value of 𝑥, the population of 𝑦-values is Normally distributed.

• There exists a straight line such that, for each value of 𝑥, the mean 𝜇𝑦 of the corresponding population of 𝑦-values lies on that straight line.

Answer explanation

There is no sample size limitation in regression.

An old saying in golf is “You drive for show and you putt for dough.” The point is that good putting is more important than long driving for shooting low scores and hence winning money. To see if this is the case, data from a random sample of 69 of the nearly 1000 players on the PGA Tour’s world money list are examined. The average number of putts per hole (fewer is better) and the player’s total winnings for the previous season are recorded and a least-squares regression line was fitted to the data. Assume the conditions for inference about the slope are met. Given is computer output from the regression analysis. The 𝑃-value is 0.0087 for the test of the hypotheses 𝐻0:𝛽=0 versus 𝐻𝑎:𝛽<0. Which of the following is a correct interpretation of this result?

• If there is no linear relationship between average number of putts per hole and total winnings for the players on the PGA Tour’s world money list, the probability of getting a random sample of 69 players that yields a least-squares regression line with a slope of –4,139,198 or less is 0.0087.

• If there is no linear relationship between average number of putts per hole and total winnings for the players in the sample, the probability of getting a random sample of 69 players that yields a least-squares regression line with a slope of –4,139,198 or less is 0.0087.

• The probability there is no linear relationship between average number of putts per hole and total winnings for these 69 players is 0.0087.

• The probability there is no linear relationship between average number of putts per hole and total winnings for all players on the PGA Tour’s world money list is 0.0087.

• The probability of making a Type I error is 0.0087.

Answer explanation

Remember that the 𝑃-value measures how likely it is to get a sample result at least as extreme as the observed result, assuming that the null hypothesis is true from the population.

An old saying in golf is “You drive for show and you putt for dough.” The point is that good putting is more important than long driving for shooting low scores and hence winning money. To see if this is the case, data from a random sample of 69 of the nearly 1000 players on the PGA Tour’s world money list are examined. The average number of putts per hole (fewer is better) and the player’s total winnings for the previous season are recorded and a least-squares regression line was fitted to the data. Assume the conditions for inference about the slope are met. Given is computer output from the regression analysis. The 𝑃-value of this test is 0.00087. Which of the following would make the 𝑃-value invalid?

• If the scatterplot of the sample data wasn't perfectly linear

• If the standard deviation of earnings is much larger than the standard deviation of putting average

• If the distribution of earnings has an outlier

• If the distribution of earnings wasn’t approximately Normal

• If the earnings for golfers with small putting averages was much more variable than the earnings for golfers with large putting averages

Answer explanation

All that is required is that the scatterplot's overall pattern is "roughly linear".

Recent revenue shortfalls in a midwestern state led to a reduction in the state budget for higher education. To offset the reduction, the largest state university proposed a 25% tuition increase. It was determined that such an increase was needed simply to compensate for the lost support from the state. Separate random samples of 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors from the university were asked whether they were strongly opposed to the increase, given that it was the minimum increase necessary to maintain the university’s budget at current levels. The results are given in the table. Which null hypothesis would be appropriate for performing a chi-square test?

• The mean number of students who are strongly opposed is the same for each of the 4 years.

• The distribution of student opinion about the proposed tuition increase is the same for each of the 4 years at this university.

• There is an association between year in school and opinion about the tuition increase at this university.

• Year in school and student opinion about the tuition increase are independent in the sample.

• The closer students get to graduation, the less likely they are to be opposed to tuition increases.

Answer explanation

The null hypothesis should contain a statement that there is no difference in the true distribution of a categorical variable two 2 or more populations or no association between two categorical variables in a single population.

Inference about the slope 𝛽 of a least-squares regression line is based on which of the following distributions?

• The standard Normal distribution

• The 𝑡 distribution with 𝑛 – 1 degrees of freedom

• The 𝑡 distribution with 𝑛 – 2 degrees of freedom

• The Normal distribution with mean 𝜇 and standard deviation 𝜎

• The chi-square distribution with 𝑛 – 1 degrees of freedom

Answer explanation

There are two variables, so the degrees of freedom is 𝑛 – 2.