inference for 2

A two-sample t-test for a difference in means was conducted to investigate whether defensive players on a football team can bench-press more weight, on average, than offensive players. The conditions for inference were met, and the test produced a test statistic of t=1.083 and a p-value of 0.15.

Based on the p-value and a significance level of �=0.05, which of the following is the correct conclusion?

• Reject the null hypothesis because 0.15>0.05. There is not convincing evidence that defensive players can bench-press more weight, on average, than offensive players.

• Reject the null hypothesis because 0.15>0.05. There is convincing evidence that defensive players can bench-press more weight, on average, than offensive players.

• Fail to reject the null hypothesis because 0.15>0.05. There is not convincing evidence that defensive players can bench-press more weight, on average, than offensive players.

• Fail to reject the null hypothesis because 0.15>0.05. There is convincing evidence that defensive players can bench-press more weight, on average, than offensive players.

• Fail to reject the null hypothesis because 0.15>0.05. There is convincing evidence that defensive players can bench-press the same amount of weight, on average, as offensive players.

To test the durability of cell phone screens, phones are dropped from a height of 1 meter until they break. A random sample of 40 phones was selected from each of two manufacturers. The phones in the samples were dropped until the screens broke. The difference in the mean number of drops was recorded and used to construct the 90 percent confidence interval (0.46,⁢1.82) to estimate the population difference in means.

Consider the sampling procedure taking place repeatedly. Each time samples are selected, the phones are dropped and the statistics are used to construct a 90 percent confidence interval for the difference in means. Which of the following statements is a correct interpretation of the intervals?

• Approximately 90 percent of the intervals will extend from 0.46 to 1.82.

• Approximately 90 percent of the intervals constructed will capture the difference in sample means.

• Approximately 90 percent of the intervals constructed will capture the difference in population means.

• Approximately 90 percent of the intervals constructed will capture at least one of the sample means.

• Approximately 90 percent of the intervals constructed will capture at least one of the population means.

A national consumer agency selected independent random samples of 45 owners of newer cars (less than five years old) and 40 owners of older cars (more than five years old) to estimate the difference in mean dollar cost of yearly routine maintenance, such as oil changes, tire rotations, filters, and wiper blades. The agency found the mean dollar cost per year for newer cars was $195 with a standard deviation of $46. For older cars, the mean was $286 with a standard deviation of $58.

Which of the following represents the 95 percent confidence interval to estimate the difference (newer minus older) in the mean dollar cost of routine maintenance between newer and older cars?

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Animal researchers studying cows and horses conducted a two-sample t-test for a difference in means to investigate whether grazing cows eat more grass, on average, than grazing horses. All conditions for inference were met, and the test produced a test statistic of t=1.664 and a p-value of 0.0487.

Which of the following is a correct interpretation of the p-value?

• The probability that cows eat more grass than horses, on average, is 0.0487.

• The probability that cows eat the same amount of grass as horses, on average, is 0.0487.

• Assuming that the mean amount of grass eaten by cows is greater than the mean amount of grass eaten by horses, the probability of observing a test statistic of at most 1.664 is 0.0487.

• Assuming that the mean amount of grass eaten by cows is equal to the mean amount of grass eaten by horses, the probability of observing a test statistic of at most 1.664 is 0.0487.

• Assuming that the mean amount of grass eaten by cows is equal to the mean amount of grass eaten by horses, the probability of observing a test statistic of at least 1.664 is 0.0487.

A study was conducted to investigate whether the mean price of a dozen eggs was different for two different grocery stores, Store A and Store B, in a large city. A carton of one dozen eggs from each store was randomly selected for each of 35 weeks, for a total sample size of 35 cartons from each store. The mean price of the 35 cartons was recorded for each store. The difference in the mean carton price for the stores will be calculated.

Which of the following is the appropriate test for the study?

• A one-sample t-test for a population proportion

• A one-sample t-test for a sample mean

• A matched-pairs t-test for a mean difference

• A two-sample t-test for a difference between population means

• A two-sample t-test for a difference between population proportions

A two-sample t-test for a difference in means was conducted to investigate whether the average time to swim a lap with the freestyle stroke is different from the average time to swim a lap with the butterfly stroke. With all conditions for inference met, the test produced a test statistic of t=−2.073 and a p-value of 0.042.

Based on the p-value and a significance level of alpha=0.05, which of the following is a correct conclusion?

• There is convincing statistical evidence that the average time to swim a lap with the freestyle stroke is less than the average time to swim a lap with the butterfly stroke.

• There is convincing statistical evidence that the average time to swim a lap with the freestyle stroke is different from the average time to swim a lap with the butterfly stroke

• There is not convincing statistical evidence that the average time to swim a lap with the freestyle stroke is greater than the average time to swim a lap with the butterfly stroke.

• There is not convincing statistical evidence that the average time to swim a lap with the freestyle stroke is different from the average time to swim a lap with the butterfly stroke.

• There is not convincing statistical evidence that the average time to swim a lap with the freestyle stroke is less than the average time to swim a lap with the butterfly stroke.

A biologist studied the frequency of croaks for frogs from two different regions. From a random sample of 32 frogs located in the northern region, the mean number of croaks per hour was 21.3, and from a random sample of 38 frogs located in the southern region, the mean number of croaks per hour was 28.9. To estimate the difference in the mean number of croaks (southern minus northern), a 95 percent confidence interval was constructed from the samples. The interval was reported as (7.1,8.1).

Which of the following claims is supported by the interval?

• All southern frogs croak more times per hour than do all northern frogs.

• The northern frogs are likely to have a greater mean number of croaks per hour than the southern frogs.

• The southern frogs are likely to have a greater mean number of croaks per hour than the northern frogs.

• All frogs in the study have about the same number of croaks per hour.

• The northern and southern frogs have the same mean number of croaks per hour.

A consumer group studied two different manufacturers of cars, J and K, to investigate differences in gas mileage for cars made by the two manufacturers. For a similar type of car, a random sample of 15 cars from J and a random sample of 12 cars from K were selected, and the gas mileages, in miles per gallon (mpg), were recorded. The difference in the sample mean gas mileages was used to construct the 90 percent confidence interval (3.5,5.7).

Assuming all conditions for inference were met, which of the following is a correct interpretation of the interval?

• The probability is 0.90 that the difference in sample means for gas mileage for the two car manufacturers is between 3.5 mpg and 5.7 mpg.

• The probability is 0.90 that the population mean difference in gas mileage for the two car manufacturers is between 3.5 mpg and 5.7 mpg.

• About 90 percent of the differences in gas mileage for the two car manufacturers are between 3.5 mpg and 5.7 mpg.

• We are 90 percent confident that the difference in sample means for gas mileage for the two car manufacturers is between 3.5 mpg and 5.7 mpg.

• We are 90 percent confident that the population mean difference of gas mileage for the two car manufacturers is between 3.5 mpg and 5.7 mpg.

A 99 percent confidence interval for a difference in means was given as 25.1±4.3.

Assuming all conditions for inference were met, which of the following is a correct interpretation of the 99 percent confidence level?

• In repeated samples of the same size, approximately 99 percent of the intervals constructed from the samples will extend from 20.8 to 29.4.

• In repeated samples of the same size, approximately 99 percent of the sample means will fall between 20.8 and 29.4.

• In repeated samples of the same size, approximately 99 percent of the samples will fall between 20.8 and 29.4.

• In repeated samples of the same size, approximately 99 percent of the intervals constructed from the samples will capture the difference in sample means.

• In repeated samples of the same size, approximately 99 percent of the intervals constructed from the samples will capture the difference in population means.

Researchers investigated whether there is a difference between two headache medications, R and S. Researchers measured the mean times required to obtain relief from a headache for patients taking one of the medications. From a random sample of 75 people with chronic headaches, 38 were randomly assigned to medication R and the remaining 37 were assigned to medication S. The time, in minutes, until each person experienced relief from a headache was recorded. The sample mean times were calculated for each medication.

Have the conditions been met for inference with a confidence interval for the difference in population means?

• Yes, all conditions have been met.

• No, because the data were not collected using a random sample.

• No, because cause and effect cannot be inferred since there is a random sample.

• No, because the sample sizes are not large enough to assume the distribution of the difference in sample means is approximately normal.

• No, because the sample sizes are not the same.