stats ch 10-11

If the consequences of making a Type I error are​ severe, would you choose the level of​ significance, alphaα​, to equal​ 0.01, 0.05, or​ 0.10?

.01

If we do not reject the null hypothesis when the statement in the alternative hypothesis is​ true, we have made a Type​ _______ error.

2

Can pleasant smells improve​ learning? Researchers timed 21 subjects as they tried to complete​ paper-and-pencil mazes. Each subject attempted a maze both with and without the presence of a floral aroma. Subjects were randomized with respect to whether they did the scented trial first or second. Suppose a paired​ t-test is to be performed to determine whether there is evidence to indicate that the time to complete the maze is faster in scented trials compared to unscented​ trials, on average. The summary statistics for the difference in time to complete the maze​ (in seconds) between the unscented and scented trials ​(unscentedminus−​scented) are x overbarxequals=3.85 and sequals=13.01. How many degrees of freedom does the​ t-statistic have?

20

Some have argued that throwing darts at the stock pages to decide which companies to invest in could be a successful​ stock-picking strategy. Suppose a researcher decides to test this theory and randomly chooses 100100 companies to invest in. After 1​ year, 5353 of the companies were considered​ winners; that​ is, they outperformed other companies in the same investment class. To assess whether the​ dart-picking strategy resulted in a majority of​ winners, the researcher tested Upper H 0H0​: pequals=0.5 versus Upper H 1H1​: pgreater than>0.5 and obtained a​ P-value of 0.27430.2743. Explain what this​ P-value means and write a conclusion for the researcher.​ (Assume alphaα is 0.1 or​ less.)

About 27 in 100 samples will give a sample proportion as high or higher than the one obtained if the population proportion really is 0.5. Because the​ P-value is​ large, do not reject the null hypothesis. There is not sufficient evidence to conclude that the​ dart-picking strategy resulted in a majority of winners.

A​ p-value is the probability​ _____________

A​ p-value is the probability of observing the actual​ result, a sample​ mean, for​ example, or something more unusual just by chance if the null hypothesis is true.

Researchers conducted a study and obtained a​ p-value of 0.75. Based on this​ p-value, what conclusion should the researchers​ draw?

Fail to reject the null hypothesis but do not accept the null hypothesis as true either.

An advertisement for a diet program described a study performed on 70 obese adults. Each person in the study was weighed before beginning the diet and then 5 weeks after starting the diet. The difference was recordeddash-a positive value indicates a person lost weight on the diet while a negative value indicates the person gained weight while on the diet. The advertisement gave a​ 95% confidence interval for the average weight change while on the​ diet: ​(minus−​1,5) lbs. The advertisement says that this shows the diet works at reducing weight for obese adults since more people lost weight than gained weight. What conclusion can be made about the weight loss​ program?

Their interpretation of the confidence interval is not correct. Because 0 is in the​ interval, there is no evidence to indicate that the average weight loss is more than 0 for obese adults.

Researchers timed 21 subjects as they tried to complete​ paper-and-pencil mazes. Each subject attempted a maze both with and without the presence of a floral aroma. Subjects were randomized with respect to which trial they did first. Suppose a paired​ t-test is to be performed to determine whether there is evidence to indicate that the time to complete the maze is faster in scented trials compared to unscented​ trials, on average. The​ p-value from the paired​ t-test is 0.11. Which of the following is the most appropriate conclusion based on this​ p-value?

There is not sufficient evidence to indicate that the individuals complete mazes faster with a floral aroma present compared to when no floral aroma is​ present, on average.

Many drivers of cars that can run on regular gas actually buy premium in the belief that they will get better gas mileage. To test that​ belief, 10 cars in a company fleet were randomly selected. All the cars run on regular gas. Each car was filled first with either regular or premium​ gasoline, decided by a coin toss. The mileage for that tank of gas was recorded. Then the car was filled with the other type of gasoline and the mileage for that tank of gas was recorded. The summary statistics for the difference in gas mileage between the two types of gasoline ​(premiumminus−​regular) for the 10 cars are x overbarxequals=2 and sequals=1.414. Suppose the​ p-value was 0.001. Which of the following is the correct​ conclusion?

There is sufficient evidence to indicate that the gas mileage using premium gasoline is better than using regular​ gasoline, on average.

A certain marathon has had a wheelchair division since 1977. An interested fan wondered who is​ faster: the​ men's marathon winner or the​ women's wheelchair marathon​ winner, on average. A paired​ t-test was​ performed, and the​ p-value was found to be 0.001. Which of the following is the correct​ conclusion?

There is sufficient evidence to indicate that the​ men's running winning time and the​ women's wheelchair winning time each year are​ different, on average.

A certain marathon has had a wheelchair division since 1977. An interested fan wondered what the difference in averaging winning times is between the​ women's wheelchair winner and the​ men's running winner. A​ 95% confidence interval for the difference in mean winning times​ (women's wheelchairminus−​men's ​running) was calculated using the paired​ t-methods: ​(minus−​20,minus−​2) minutes. Which of the following is the correct interpretation of this confidence​ interval?

We are​ 95% confident that​ men's winning running times are between 2 and 20 minutes slower than​ women's winning wheelchair​ times, on average.

Rejecting the null hypothesis when the null hypothesis is true is called

a Type I Error.

Sample evidence can prove that a null hypothesis is true.

false

Can pleasant smells improve​ learning? Researchers timed 21 subjects as they tried to complete​ paper-and-pencil mazes. Each subject attempted a maze both with and without the presence of a floral aroma. Subjects were randomized with respect to whether they did the scented trial first or second and the differences between the unscented and scented times were computed ​(unscentedminus−​scented). Suppose a paired​ t-test is to be performed to determine whether there is evidence to indicate that the time to complete the maze is faster in scented trials compared to unscented​ trials, on average. Which of the following is the correct null​ hypothesis?

mu Subscript diffμdiffequals=​0, where mu Subscript diffμdiff is the mean difference between the time taken to complete the maze in unscented and scented trials

In​ 1993, the British Medical Journal published an article​ titled, "Is Friday the 13th Bad for Your​ Health?" Researchers in Britain examined how Friday the 13th affects human behavior. One question was whether people tend to stay at home more on Friday the 13th. The accompanying data give the number of cars passing Junctions 9 and 10 on the M25 motorway for consecutive Fridays​ (the 6th and​ 13th) for five different time periods. Assuming all conditions for inference are​ met, which test is appropriate to use to answer the​ researcher's question of​ interest?

paired​ t-test

Some students do homework with the TV on. Researchers wanted to see if people can work as effectively with distractions as without them. From a group of student​ volunteers, researchers randomly assigned half to do a relatively easy crossword puzzle in a quiet room and the other half to do the same crossword puzzle in a room in which a TV was on. Then each volunteer was to do a different relatively easy crossword puzzle under the other condition. Students were timed on how long it took to complete the puzzle and the difference in each​ student's quiet-room time and​ TV-room time was recorded. Which test is appropriate to use to determine if students take longer to complete the puzzle in the TV room compared to the quiet​ room, on​ average?

paired​ t-test

The null and alternative hypotheses are given. Determine whether the hypothesis test is​ left-tailed, right-tailed, or​ two-tailed. What parameter is being​ tested?

right tailed; population proportion

A professor wondered if there was a difference in the proportion of students who dropped math classes between females and males. The professor randomly selected 20 math classes around campus and recorded the gender of the individual and whether or not a student enrolled in the class at the beginning of the term dropped the class at some point during the term. Assuming all conditions are​ satisfied, which of the following tests should the researcher​ use?

two-sample z-test for proportions

Can pleasant smells improve​ learning? Researchers timed 21 subjects as they tried to complete​ paper-and-pencil mazes. Each subject attempted a maze both with and without the presence of a floral aroma. Subjects were randomized with respect to whether they did the scented trial first or second and the differences between the unscented and scented times were computed ​(unscentedminus−​scented). Suppose a paired​ t-test is to be performed to determine whether there is evidence to indicate that the time to complete the maze is faster in scented trials compared to unscented​ trials, on average. Which of the following is the correct alternative​ hypothesis?

​: mu Subscript diffμdiffgreater than>​0, where mu Subscript diffμdiff is the mean difference between the time taken to complete the maze in unscented and scented trials