Statistics Confidence Intervals and Significance Test

At a baseball game, 42 of 65 randomly selected people own an iPod. At a rock concert occurring at the same time across town, 34 of 52 randomly selected people own an iPod. A researcher wants to test the claim that the proportion of iPod owners at the two events is different. A 90% confidence interval for the difference (Game - Concert) in populations proportions is (-0.154, 0.138). Which of the following gives the correct outcome of the researcher's test of the claim?

• Because the interval includes 0, the researcher can conclude that the proportion of iPod owners at the two events is the same.

• Because the center of the interval is -0.008, the researcher can conclude that a higher proportion of people at the rock concert own iPods than at the baseball game.

• Because the interval includes 0, the researcher cannot conclude that the proportion of iPod owners at the two events is different.

• Because the interval includes -0.008, the researcher cannot conclude that the proportion of iPod owners at the two events is different.

• Because the interval includes more negative than positive values, the researcher can conclude that a higher proportion of people at the rock concert own iPods than at the baseball game.

• 0.90

• 0.40

• 0.05

• 0.20

• 0.10

Conference organizers wondered whether posting a sign that says "Please take only one cookie" would reduce the proportion of conference attendees who take multiple cookies from the snack table during a break. To find out the organizers randomly assigned 212 attendees to take their break in a room where the snack table had a sign posted, and 189 attendees were assigned to a room where the snack table did not have a sign posted. In the room without a sign posted, 24.3% of attendees took multiple cookies. In the room with the sign posted, 17% of attendees took multiple cookies. Is this decrease in proportions statistically significant at the 5% level?

• No, the P-value is 0.034

• No, the P-value is 0.068

• Yes, the P-value is 0.034

• Yes, The P-value si 0.068

• Cannot be determined from the information given.

What is the critical z-score for a 94% confidence interval?

• 2.57

• 2.33

• 1.95

• 1.88

• 1.75

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 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 5% , 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.

• B) 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.

• C) 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.

• D) 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.

• E) 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.

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.

The director of fitness for a large corporation with over 5,000 employees recorded the resting heart rate, in beats per minute (bpm) , for 35 employees who were known to wear activity trackers. The following boxplot summarizes the result.

 

The director wants to estimate the resting heart rate for all employees with a confidence interval. Have all conditions for inference been met?

• Yes, all conditions have been met.

• No, the distribution of the sample data is not approximately symmetric.

• No, the sample size is greater than 10 percent of the population size.

• No, the distribution of resting heart rate in the population cannot be assumed to be approximately normal.

• No, the sample was not selected at random

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.

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