Q&A Hero

Q: If you are interested in estimating the difference between the means of two samples that have been paired, the point estimate for this difference is the mean value of the paired differences.

Q: If the sample data lead us to suspect that the variances of the two populations are not equal, the t-test statistic and the degrees of freedom must be adjusted accordingly.

Q: In order to test the difference in populations means, samples were collected for two independent populations where the variances are assumed equal and the population normally distributed. The following data resulted: Population 1 Population 2 = 112 = 107 s = 14 s = 17 n = 25 n = 28 The value of the pooled standard deviation is 15.66.

Q: The Sergio Lumber Company manufactures plywood. One step in the process is the one where the veneer is dried by passing through a huge dryer (similar to an oven) where much of the moisture in the veneer is extracted. At the end of this step, samples of veneer are tested for moisture content. It is believed that pine veneer will be less moist on average than will fir veneer. The following data were reported recently where the values represent the percent moisture in the wood: Pine Fir 13.1 19.8 10.2 13.5 14.7 20.3 17.8 20.5 17.6 21.4 19.2 23.4 7.4 14.6 13.3 18.8 17.3 10.7 Based on these data, the critical t value from the t-distribution will be 1.7459 if the significance level is set at 0.05 and variances are presumed equal.

Q: To find the pooled standard deviation involves taking a weighted average of the two sample variances, then finding its square root.

Q: When performing a hypothesis test for the difference between the means of two independent populations where the standard deviations are known, it is necessary to use the pooled standard deviation in calculating the test statistic.

Q: The t-distribution can be used to test hypotheses about the difference between two population means given the following two assumptions: - each population is normally distributed, and - the two populations have equal variances.

Q: Two samples are said to be independent if they are collected at different points in time.

Q: In order to make the test for the difference between two population means valid, the sample size in each independent sample must be the same.

Q: The t-distribution is still applicable even when there are small violations of the assumptions for the case when the variances for two populations are unknown. This is particularly true when the sample sizes are approximately equal.

Q: All other things held constant, increasing the level of confidence for a confidence interval estimate for the difference between two population means will result in a wider confidence interval estimate.

Q: Increasing the size of the samples in a study to estimate the difference between two population means will increase the level of confidence that a decision maker can have regarding the interval estimate.

Q: Recently the managers for a large retail department store stated that a study has revealed that female shoppers spend on average 23.5 minutes longer in the store per visit than do male shoppers. Based on this information, the managers can be confident that female shoppers, as a population, do spend longer times in the store than do males shoppers, as a population.

Q: In estimating a confidence interval for the difference between two means, when the samples are independent and the standard deviations are unknown, it can be acceptable for there to be small violations of the assumptions of normality and equal variances, especially when the sample sizes are equal.

Q: Box and whisker plots are often useful for determining whether two populations have distributions that might each be normally distributed.

Q: In estimating the difference between two population means based on small, independent samples from the two populations, two important assumptions are that the populations each be normally distributed and the populations have equal variances.

Q: The NCAA is interested in estimating the difference in mean number of daily training hours for men and women athletes on college campuses. It wants 95 percent confidence and will select a sample of 10 men and 10 women for the study. The variances are assumed equal and the populations normally distributed. The sample results are: Men Women n1= 10 students n2= 10 students 1= 2.7 hours 2= 2.4 hours s1= .30 hours s2= .40 hours Based on these data, the lower limit for the difference between population means is 0.15 hours.

Q: In estimating the difference between two population means, if a 95 percent confidence interval includes zero, then we can conclude that there is a 95 percent chance that the difference between the two population means is zero.

Q: The Cranston Hardware Company is interested in estimating the difference in the mean purchase for men customers versus women customers. It wishes to estimate this difference using a 95 percent confidence level. Assume that the variances are equal and the populations normally distributed. The following data represent independent samples from each population: Men Women 16.49 17.21 33.34 17.46 20.18 15.65 26.39 10.67 28.03 14.07 26.02 19.61 16.08 15.90 32.27 11.17 21.66 24.66 32.32 13.35 29.79 20.87 33.37 22.57 Based on these data, the company can conclude that there is a statistical difference between men and women with regard to mean spending at the hardware store with men tending to spend more on average than women.

Q: To find a confidence interval for the difference between the means of independent samples, when the variances are unknown but assumed equal, the sample sizes of the two groups must be the same.

Q: A consumer group plans to test whether a new passenger car that is advertised to have a mean highway miles per gallon of at least 33 actually meets this level. They plan to test the hypothesis using a significance level of 0.05 and a sample size of n = 100 cars. It is believed that the population standard deviation is 3 mpg. Based upon this information, what is the critical value in terms of miles per gallon that would be needed prior to finding beta? A) 32.5065 B) 33.4935 C) 33.588 D) 32.412

Q: Which of the following will be helpful if the decision maker wishes to reduce the chance of making a Type II error? A) Increase the level of significance at which the hypothesis test is conducted. B) Increase the sample size. C) Both A and B will work. D) Neither A nor B will be effective.

Q: The power of a test is measured by its capability of: A) rejecting a null hypothesis that is true. B) not rejecting a null hypothesis that is true. C) rejecting a null hypothesis that is false. D) not rejecting a null hypothesis that is false.

Q: For a given sample size n, if the level of significance (α) is decreased, the power of the test: A) will increase. B) will decrease. C) will remain the same. D) cannot be determined.

Q: If the Type I error (α) for a given test is to be decreased, then for a fixed sample size n: A) the Type II error (β) will also decrease. B) the Type II error (β) will increase. C) the power of the test will increase. D) a one-tailed test must be utilized.

Q: Which of the following is not a required step in finding beta? A) Assuming a true value of the population parameter where the null is false B) Finding the critical value based on the null hypothesis C) Converting the critical value from the standard normal distribution to the units of the data D) Finding the power of the test

Q: Mike runs for the president of the student government and is interested to know whether the proportion of the student body in favor of him is significantly more than 50 percent. A random sample of 100 students was taken. Fifty-five of them favored Mike. At a 0.05 level of significance, it can be concluded that the proportion of the students in favor of Mike A) is significantly greater than 50 percent because 55 percent of the sample favored him. B) is not significantly greater than 50 percent. C) is significantly greater than 55 percent. D) is not significantly different from 55 percent.

Q: After completing sales training for a large company, it is expected that the salesperson will generate a sale on at least 15 percent of the calls he or she makes. To make sure that the sales training process is working, a random sample of n = 400 sales calls made by sales representatives who have completed the training have been selected and the null hypothesis is to be tested at 0.05 alpha level. Suppose that a sale is made on 36 of the calls. Based on these sample data, which of the following is true? A) The null hypothesis should be rejected since the test statistic falls in the lower tail rejection region. B) The null hypothesis is supported since the sample results do not fall in the rejection region. C) There is insufficient evidence to reject the null hypothesis and the sample proportion is different from the hypothesized proportion due to sampling error. D) It is possible that a Type II statistical error has been committed.

Q: Woof Chow Dog Food Company believes that it has a market share of 25 percent. It surveys n = 100 dog owners and ask whether or not Woof Chow is their regular brand of dog food, and 23 people say yes. Based upon this information, what is the value of the test statistic? A) -0.462 B) -0.475 C) 0.462 D) 0.475

Q: The manager of an online shop wants to determine whether the mean length of calling time of its customers is significantly more than 3 minutes. A random sample of 100 customers was taken. The average length of calling time in the sample was 3.1 minutes with a standard deviation of 0.5 minutes. At a 0.05 level of significance, it can be concluded that the mean of the population is: A) significantly greater than 3. B) not significantly greater than 3. C) significantly less than 3. D) not significantly different from 3.10.

Q: A house cleaning service claims that it can clean a four bedroom house in less than 2 hours. A sample of n = 16 houses is taken and the sample mean is found to be 1.97 hours and the sample standard deviation is found to be 0.1 hours. Using a 0.05 level of significance the correct conclusion is: A) reject the null because the test statistic (-1.2) is < the critical value (1.7531). B) do not reject the null because the test statistic (1.2) is > the critical value (-1.7531). C) reject the null because the test statistic (-1.7531) is < the critical value (-1.2). D) do not reject the null because the test statistic (-1.2) is > the critical value (-1.7531).

Q: A major airline is concerned that the waiting time for customers at its ticket counter may be exceeding its target average of 190 seconds. To test this, the company has selected a random sample of 100 customers and times them from when the customer first arrives at the checkout line until he or she is at the counter being served by the ticket agent. The mean time for this sample was 202 seconds with a standard deviation of 28 seconds. Given this information and the desire to conduct the test using an alpha level of 0.02, which of the following statements is true? A) The chance of a Type II error is 1 - 0.02 = 0.98. B) The test to be conducted will be structured as a two-tailed test. C) The test statistic will be approximately t = 4.286, so the null hypothesis should be rejected. D) The sample data indicate that the difference between the sample mean and the hypothesized population mean should be attributed only to sampling error.

Q: When testing a two-tailed hypothesis using a significance level of 0.05, a sample size of n = 16, and with the population standard deviation unknown, which of the following is true? A) The null hypothesis can be rejected if the sample mean gets too large or too small compared with the hypothesized mean. B) The alpha probability must be split in half and a rejection region must be formed on both sides of the sampling distribution. C) The test statistic will be a t-value. D) All of the above are true.

Q: A concern of Major League Baseball is that games last too long. Some executives in the league's headquarters believe that the mean length of games this past year exceeded 3 hours (180 minutes). To test this, the league selected a random sample of 80 games and found the following results: = 193 minutes and s = 16 minutes. Based on these results, if the null hypothesis is tested using an alpha level equal to 0.10, which of the following is true? A) The null hypothesis should be rejected if > 182.31. B) The test statistic is t = 1.2924. C) Based on the sample data, the null hypothesis cannot be rejected. D) It is possible that when the hypothesis test is completed, a Type II statistical error has been made.

Q: The R.D. Wilson Company makes a soft drink dispensing machine that allows customers to get soft drinks from the machine in a cup with ice. When the machine is running properly, the average number of fluid ounces in the cup should be 14. Periodically the machines need to be tested to make sure that they have not gone out of adjustment. To do this, six cups are filled by the machine and a technician carefully measures the volume in each cup. In one such test, the following data were observed: 14.25 13.7 14.02 14.13 13.99 14.04 Based on these sample data, which of the following is true if the significance level is .05? A) No conclusion can be reached about the status of the machine based on a sample size of only six cups. B) The null hypothesis cannot be rejected since the test statistic is approximately t = 0.20, which is not in the rejection region. C) The null hypothesis can be rejected since the sample mean is greater than 14. D) The null can be rejected because the majority of the sample values exceed 14.

Q: The cost of a college education has increased at a much faster rate than costs in general over the past twenty years. In order to compensate for this, many students work part- or full-time in addition to attending classes. At one university, it is believed that the average hours students work per week exceeds 20. To test this at a significance level of 0.05, a random sample of n = 20 students was selected and the following values were observed: 26 15 10 40 10 20 30 36 40 0 5 10 20 32 16 12 40 36 10 0 Based on these sample data, the critical value expressed in hours: A) is approximately equal to 25.26 hours. B) is approximately equal to 25.0 hours. C) cannot be determined without knowing the population standard deviation. D) is approximately 22 hours.

Q: A company that makes shampoo wants to test whether the average amount of shampoo per bottle is 16 ounces. The standard deviation is known to be 0.20 ounces. Assuming that the hypothesis test is to be performed using 0.10 level of significance and a random sample of n = 64 bottles, how large could the sample mean be before they would reject the null hypothesis? A) 16.2 ounces B) 16.049 ounces C) 15.8 ounces D) 16.041 ounces

Q: The reason for using the t-distribution in a hypothesis test about the population mean is: A) the population standard deviation is unknown. B) it results in a lower probability of a Type I error occurring. C) it provides a smaller critical value than the standard normal distribution for a given sample size. D) the population is not normally distributed.

Q: In a two-tailed hypothesis test for a population mean, an increase in the sample size will: A) have no effect on whether the null hypothesis is true or false. B) have no effect on the significance level for the test. C) result in a sampling distribution that has less variability. D) All of the above are true.

Q: A hypothesis test is to be conducted using an alpha = .05 level. This means: A) there is a 5 percent chance that the null hypothesis is true. B) there is a 5 percent chance that the alternative hypothesis is true. C) there is a maximum 5 percent chance that a true null hypothesis will be rejected. D) there is a 5 percent chance that a Type II error has been committed.

Q: If the p value is less than α in a two-tailed test, A) the null hypothesis should not be rejected. B) the null hypothesis should be rejected. C) a one-tailed test should be used. D) More information is needed to reach a conclusion about the null hypothesis.

Q: If an economist wishes to determine whether there is evidence that average family income in a community exceeds $25,000. The best null hypothesis is: A) μ = 25,000. B) μ > 25,000. C) μ ≤ 25,000. D) μ ≥ 25,000.

Q: If we are performing a two-tailed test of whether μ = 100, the probability of detecting a shift of the mean to 105 will be ________ the probability of detecting a shift of the mean to 110. A) less than B) greater than C) equal to D) not comparable to

Q: Which of the following would be an appropriate null hypothesis? A) The mean of a population is equal to 55. B) The mean of a sample is equal to 55. C) The mean of a population is greater than 55. D) The mean of a sample is greater than 55.

Q: In a hypothesis test involving a population mean, which of the following would be an acceptable formulation? A) H0: ≤ $1,700 Ha: > $1,700 B) H0: > $1,700 Ha: ≥ $1,700 C) H0: μ ≤ $1,700 Ha: μ > $1,700 D) None of the above is a correct formulation.

Q: Which of the following statements is true? A) The decision maker controls the probability of making a Type I statistical error. B) Alpha represents the probability of making a Type II error. C) Alpha and beta are directly related such that when one is increased the other will increase also. D) The alternative hypothesis should contain the equality.

Q: When someone is on trial for suspicion of committing a crime, the hypotheses are: H0: innocent HA: guilty Which of the following is correct? A) Type I error is acquitting a guilty person. B) Type I error is convicting an innocent person. C) Type II error is acquitting an innocent person. D) Type II error is convicting an innocent person.

Q: A major airline has stated in an industry report that its mean onground time between domestic flights is less than 18 minutes. To test this, the company plans to sample 36 randomly selected flights and use a significance level of .10. Assuming that the population standard deviation is known to be 4.0 minutes, if the true population mean is 16 minutes, the decision maker could end up making either a Type I or a Type II error depending on the sample result.

Q: The director of the city Park and Recreation Department claims that the mean distance people travel to the city's greenbelt is more than 5.0 miles. Assuming that the population standard deviation is known to be 1.2 miles and the significance level to be used to test the hypothesis is 0.05 when a sample size of n = 64 people are surveyed, the probability of a Type II error is approximately .4545 when the "true" population mean is 5.5 miles.

Q: A city newspaper has stated that the average time required to sell a used car advertised in the paper is less than 5 days. Assuming that the population standard deviation is 2.1 days, if the "true" population mean is 4.1 days and a sample size of n = 49 is used with an alpha equal to 0.05, the probability that the hypothesis test will lead to a Type II error is approximately .0869.

Q: The probability of a Type II error decreases as the "true" population value gets farther from the hypothesized population value, given that everything else is held constant.

Q: To calculate beta requires making a "what if" assumption about the true population parameter, where the "what-if" value is one that would cause the null hypothesis to be false.

Q: If a decision maker wishes to reduce the chance of making a Type II error, one option is to increase the sample size.

Q: Type II errors are typically greater for two-tailed hypothesis tests than for one-tailed tests.

Q: If a decision maker is concerned that the chance of making a Type II error is too large, one option that will help reduce the risk is to reduce the significance level.

Q: Choosing an alpha of 0.01 will cause beta to equal 0.99.

Q: The chance of making a Type II statistical error increases if the "true" population mean is closer to the hypothesized population mean, all other factors held constant.

Q: An article in an operations management journal recently stated that a formal hypothesis test rejected the hypothesis that mean employee productivity was less than $45.70 per hour in the wood processing industry. Given this conclusion, it is possible that a Type I statistical error was committed.

Q: In a hypothesis test, increasing the sample size will generally result in a smaller chance of making a Type I error since sampling error is likely to be reduced.

Q: One claim states the IRS conducts audits for not more than 5 percent of total tax returns each year. In order to test this claim statistically, the appropriate null and alternative hypotheses are: H0: μ ≤ 0.05 Ha: μ > 0.05

Q: A cell phone company believes that 90 percent of their customers are satisfied. They survey a sample of n = 100 customers and find that 82 say they are satisfied. In calculating the standard error of the sampling distribution (σp) the proportion to use is 0.82.

Q: A major package delivery company claims that at least 95 percent of the packages it delivers reach the destination on time. As part of the evidence in a lawsuit against the package company, a random sample of n = 200 packages was selected. A total of 188 of these packages were delivered on time. Using a significance level of .05, the test statistic for this test is approximately z = -0.65.

Q: When the hypothesized proportion is close to 0.50, the spread in the sampling distribution of is greater than when the hypothesized proportion is close to 0.0 or 1.0.

Q: When testing a hypothesis involving population proportions, an increase in sample size will result in a smaller chance of making a Type I statistical error.

Q: Aceco has a contract with a supplier to ship parts that contain no more than three percent defects. When a large shipment of parts comes in, Aceco samples n = 150. Based on the results of the sample, they either accept the shipment or reject it. If Aceco wants no more than a 0.10 chance of rejecting a good shipment, the cut-off between accepting and rejecting should be 0.0478 or 4.78 percent of the sample.

Q: A cell phone company believes that 90 percent of its customers are satisfied with their service. They survey n = 30 customers. Based on this, it is acceptable to assume the sample distribution is normally distributed.

Q: A company that makes and markets a device that is aimed at helping people quit smoking claims that at least 70 percent of the people who have used the product have quit smoking. To test this, a random sample of n = 100 product users was selected. The critical value for the hypothesis test using a significance level of 0.05 would be approximately -1.645.

Q: The executive director of the United Way believes that more than 24 percent of the employees in the high-tech industry have made voluntary contributions to the United Way. In order to test this statistically, the appropriate null and alternative hypotheses are: H0: ≤ .24 HA: > .24

Q: When deciding the null and alternative hypotheses, the rule of thumb is that if the claim contains the equality (e.g., at least, at most, no different from, etc.), the claim becomes the null hypothesis. If the claim does not contain the equality (e.g., less than, more than, different from), the claim is the alternative hypothesis.

Q: For testing a research hypothesis, the burden of proof that a new product is no better than the original is placed on the new product, and the research hypothesis is formulated as the null hypothesis.

Q: Lube-Tech is a major chain whose primary business is performing lube and oil changes for passenger vehicles. The national operations manager has stated in an industry newsletter that the mean number of miles between oil changes for all passenger cars exceeds 4,200 miles. To test this, an industry group has selected a random sample of 100 vehicles that have come into a lube shop and determined the number of miles since the last oil change and lube. The sample mean was 4,278 and the standard deviation was known to be 780 miles. Based on this information, the p-value for the hypothesis test is less than 0.10.

Q: When using the p-value method for a two-tailed hypothesis, the p-value is found by finding the area in the tail beyond the test statistic, then doubling it.

Q: A two-tailed hypothesis test is used when the null hypothesis looks like the following: H0 : = 100.

Q: Generally, it is possible to appropriately test a null and alternative hypotheses using the test statistic approach and reach a different conclusion than would be reached if the p-value approach were used.

Q: When using the p-value method, the null hypothesis is rejected when the calculated p-value > α.

Q: The state insurance commissioner believes that the mean automobile insurance claim filed in her state exceeds $1,700. To test this claim, the agency has selected a random sample of 20 claims and found a sample mean equal to $1,733 and a sample standard deviation equal to $400. They plan to conduct the test using a 0.05 significance level. Based on this, the null hypothesis should be rejected if > $1,854.66 approximately.

Q: A two-tailed hypothesis test with α = 0.05 is similar to a 95 percent confidence interval.

Q: The loan manager for State Bank and Trust has claimed that the mean loan balance on outstanding loans at the bank is over $14,500. To test this at a significance level of 0.05, a random sample of n = 100 loan accounts is selected. Assuming that the population standard deviation is known to be $3,000, the null and alternative hypotheses to be tested are: H0: μ ≤ $14,500 HA: μ > $14,500

Q: The critical value in a null hypothesis test is called alpha.

Q: Type II error is failing to reject the null hypothesis when the null is actually false.