Testing a Claim

A researcher claims to have found a drug that causes people to grow taller. The coach of the basketball team at Brandon University has expressed interest but demands evidence. Over $1000$ Brandon students volunteer to participate in an experiment to test this new drug. Fifty of the volunteers are randomly selected, their heights are measured, and they are given the drug. Two weeks later, their heights are measured again. The power of the test to detect an average increase in height of one inch could be increased by 

(a) using only volunteers from the basketball team in the experiment. 

(b) using $A=0.01$ instead of $A=0.05$. 

(c) using $A=0.05$ instead of $A=0.01$. 

(d) giving the drug to $25$ randomly selected students instead of $50$. 

(e) using a two-sided test instead of a one-sided test. 

A $95\%$ confidence interval for a population mean  is calculated to be $(1.7, 3.5)$. Assume that the conditions for performing inference are met. What conclusion can we draw for a test of $H_0 : \mu =2$ versus $Ha :\mu \ne 2$ at the $A=0.05$ level based on the confidence interval?

(a) None. We cannot carry out the test without the original data.

(b) None. We cannot draw a conclusion at the $A=0.05$ level since this test is connected to the $97.5\%$ confidence interval.

(c) None. Confidence intervals and significance tests are unrelated procedures. 

(d) We would reject $H_0$ at level $A=0.05$.

(e) We would fail to reject $H_0$ at level $A=0.05$.

In a test of $H_0 :p=0.4$ against $H_a :p\ne 0.4$, a random sample of size $100$ yields a test statistic of $z=1.28$. The P-value of the test is approximately equal to 

(a) $0.90$.

(b) $0.40$. 

(c) $0.05$. 

(d) $0.20$. 

(e) $0.10$.

An SRS of 100 postal employees found that the average time these employees had worked at the postal service was $7$ years with standard deviation $2$ years. Do these data provide convincing evidence that the mean time of employment M for the population of postal employees has changed from the value of $7.5$  that was true $20$ years ago? To determine this, we test the hypotheses $H_0 : \mu =7.5$ versus $H_a : \mu \ne 7.5$ using a one-sample $t$ test. What conclusion should we draw at the $5\%$ significance level?

(a) There is convincing evidence that the mean time working with the postal service has changed. 

(b) There is not convincing evidence that the mean time working with the postal service has changed. 

(c) There is convincing evidence that the mean time working with the postal service is still $7.5$ years. 

(d) There is convincing evidence that the mean time working with the postal service is now $7$ years.

(e) We cannot draw a conclusion at the $5\%$ significance level. The sample size is too small.

Are TV commercials louder than their surrounding programs? To find out, researchers collected data on $50$ randomly selected commercials in a given week. With the television’s volume at a fixed setting, they measured the maximum loudness of each commercial and the maximum loudness in the first $30$ seconds of regular programming that followed. Assuming conditions for inference are met, the most appropriate method for answering the question of interest is

(a) a one-proportion z test. 

(b) a one-proportion z interval. 

(c) a paired t test. 

A software company is trying to decide whether to produce an upgrade of one of its programs. Customers would have to pay $\backslash (100$ for the upgrade. For the upgrade to be profitable, the company needs to sell it to more than $20\%$ of their customers. You contact a random sample of $60$ customers and find that 16 would be willing to pay $\backslash )100$ for the upgrade.

(a) Do the sample data give good evidence that more than $20\%$ of the company’s customers are willing to purchase the upgrade? Carry out an appropriate test at the $A=0.05$ significance level.

(b) Which would be a more serious mistake in this setting—a Type I error or a Type II error? Justify your answer. 

(c) Other than increasing the sample size, describe one way to increase the power of the test in (a). 

“I can’t get through my day without coffee” is a common statement from many students. Assumed benefits include keeping students awake during lectures and making them more alert for exams and tests. Students in a statistics class designed an experiment to measure memory retention with and without drinking a cup of coffee one hour before a test. This experiment took place on two different days in the same week (Monday and Wednesday). Ten students were used. Each student received no coffee or one cup of coffee, one hour before the test on a particular day. The test consisted of a series of words flashed on a screen, after which the student had to write down as many of the words as possible. On the other day, each student received a different amount of coffee (none or one cup). (a) One of the researchers suggested that all the subjects in the experiment drink no coffee before Monday’s test and one cup of coffee before Wednesday’s test. Explain to the researcher why this is a bad idea and suggest a better method of deciding when each subject receives the two treatments.

(b) The data from the experiment are provided in the table below. Set up and carry out an appropriate test to determine whether there is convincing evidence that drinking coffee improves memory.

A government report says that the average amount of money spent per U.S. household per week on food is about $\backslash (158$. A random sample of $50$ households in a small city is selected, and their weekly spending on food is recorded. The Minitab output below shows the results of requesting a confidence interval for the population mean M. An examination of the data reveals no outliers.

(a) Explain why the Normal condition is met in this case. 

(b) Can you conclude that the mean weekly spending on food in this city differs from the national figure of $\backslash )158$? Give appropriate evidence to support your answer.