BA 3 - Hypothesis Testing
What is the significance level for a 95% confidence level?
Significance level=1-confidence level. 1-0.95=0.05, that is, 5%.
If the two-sided p-value of a given sample is 0.0020, what is the one-sided p-value for that sample mean?
0.0010 is correct The one-sided p-value is half of the two-sided p-value. Thus, the one-sided p-value is 0.0020/2=0.0010.
How would you interpret the p-value of 0.0026?
If the null hypothesis is true, the likelihood of obtaining a sample with a mean at least as extreme as 7.3 is 0.26% is correct The p-value of 0.0026 indicates that if the population mean were actually still 6.7, there would be a very small possibility, just 0.26%, of obtaining a sample with a mean at least as extreme as 7.3. Equivalently, since 7.3-6.7=0.6, this p-value tells us that if the null hypothesis is true, the probability of obtaining a sample with a mean less than 6.7-0.6=6.1 or greater than 6.7+0.6=7.3 is 0.26%.
If we specify a 75% confidence level, what percentage of sample means do we expect to fall in the rejection region?
The significance level equals the area of the rejection region. The significance level equals 1-confidence level. In this case, 1-0.75=0.25, that is, 25%.
Suppose we wanted to calculate a 90% range of likely sample means for the movie theater example but our sample size had been only 15. (Assume the same historical population mean, sample mean, and sample standard deviation.) Select the function that would correctly calculate this range.
6.7±CONFIDENCE.T(0.10,2.8,15) is correct The range of likely sample means is centered at the historical population mean, in this case 6.7. We must use CONFIDENCE.T since the sample size is less than 30.
Suppose the movie theater manager had gathered a sample that had an average customer satisfaction rating of 7.05. For the two-sided test with H0:μ=6.7and Ha:μ≠6.7, the p-value is approximately 0.07. Would you reject or fail to reject the null hypothesis, μ=6.7, at the 5% significance level?
Fail to reject the null hypothesis is correct Because the p-value, 0.07, is greater than the significance level, 0.05, we do not have enough evidence to reject the null hypothesis, so we would fail to reject it.
An automotive manufacturer has developed a new type of tire that the research team believes to increase fuel efficiency. The manufacturer wants to test if there is an increase in the mean gas mileage of mid-sized sedans that use the new type of tire, compared to 32 miles per gallon, the historic mean gas mileage of mid-sized sedans not using the new tires.
The automotive manufacturer should perform a one-sided hypothesis test to analyze a change in a single population The manufacturer believes that the new tires change fuel efficiency in a single direction (i.e., that efficiency increases) and thus should use a one-sided hypothesis test. The automotive manufacturer is analyzing the change of a single population mean compared to the known historic population mean of gas mileage in mid-sized sedans.
Before beginning a hypothesis test, an analyst specified a significance level of 0.10. Which of the following is true?
There is a 10% chance of rejecting the null hypothesis when it is actually true. is correct Correct. The significance level specifies how different the observed sample mean has to be from the mean expected under the null hypothesis before we reject the null hypothesis. A significance level of 0.10 means that the observed sample mean is so different from the mean expected under the null hypothesis that it would only occur 10% of the time if the null hypothesis were true.
A car manufacturing executive introduces a new method to install a car's brakes that is much faster than the previous method. He needs to test whether the brakes installed with the new method are as safe and effective as those installed with the previous method. His null hypothesis is that the brakes installed using the new method are as safe as those installed using the old method. In this situation, would it be worse to make a type I error or a type II error?
Type II is correct A type II error, or false negative, would be that the brakes are actually not safe but the manufacturer deems them safe and proceeds with the new installation method. This would be worse than returning to the slower method, because the unsafe cars could cause injuries or fatal accidents.
A movie theater manager wants to determine whether popcorn sales have increased since the theater switched from using "butter-flavored topping" to real butter. Historically the average popcorn revenue per weekend day was approximately $3,500. After the theater started using real butter, the manager randomly sampled 12 weekend days and calculated the sample's summary statistics. The average revenue per weekend day in the sample was approximately $4,200 with a standard deviation of $140. Select the function that would correctly calculate the 90% range of likely sample means.
3,500±CONFIDENCE.T(0.10,140,12) is correct Correct. The range of likely sample means is centered at the historical population mean, in this case $3,500. Because the sample contains fewer than 30 data points, we use CONFIDENCE.T. Excel's CONFIDENCE.T function syntax is CONFIDENCE.T(alpha, standard_dev, size). Because we wish to construct a 90% range of likely sample means, alpha equals 0.10.
Suppose we wanted to calculate a 90% range of likely sample means for the movie theater example. Select the function that would correctly calculate this range.
6.7±CONFIDENCE.NORM(0.10,2.8,196) is correct The range of likely sample means is centered at the historical population mean, in this case 6.7. Since this is a 90% range of likely sample means, alpha equals 0.10.
If you are performing a hypothesis test based on a 0.10 significance level (10%), what are your chances of making a type I error?
10% is correct The probability of a type I error is equal to the significance level (which is 1-confidence level). A 10% significance level indicates that there is a 10% chance of making a type I error.
If the two-sided p-value of a given sample mean is 0.0040, what is the one-sided p-value for that sample mean?
The one-sided p-value is half of the two-sided p-value. Since the two-sided p-value is 0.0040, the one-sided p-value is 0.0040/2=0.0020.
If the one-sided p-value of a given sample mean is 0.0150, what is the two-sided p-value for that sample mean?
The two-sided p-value is double the one-sided p-value. Since the one-sided p-value is 0.0150, the two-sided p-value is 0.0150*2=0.0300.
A manager of a factory wants to know if a new quality check protocol has decreased the number of units a worker produces in a day. Before the new protocol, a worker could produce 27 units per day. What null hypothesis should the manager use to test this claim?
µ ≥ 27 units is correct This is the null hypothesis. Remember that the null and alternative hypotheses are opposites.
The mean score on a particular standardized test is 500, with a standard deviation of 100. To assess whether a training course has been effective in improving scores on the test, we take a random sample of 100 students from the course and find that the average score of this sample is 550. Which function would correctly calculate the 95% range of likely sample means under the null hypothesis?
500 ± CONFIDENCE.NORM(0.05,100,100) is correct The range of likely sample means is centered at the historical population mean, 500. Because our sample is larger than 30, we can assume the distribution of sample means is roughly normal, due to the central limit theorem, and use the CONFIDENCE.NORM function.
Suppose the average satisfaction rating of the sample is 7.0 out of 10. Which of the following do you think would be the correct conclusion? Remember that H0:μ=6.7 and Ha:μ≠6.7
Do not reject the null hypothesis is correct Although we can't be completely sure without doing the analysis, it would probably not be that unusual to draw a sample that has a mean of 7.0 if the average customer satisfaction rating has not changed, and is still 6.7. Therefore, we would probably fail to reject the null hypothesis. To be certain whether this is the case, we would have to complete the hypothesis test—that is, construct the range around the historical population mean and see whether or not 7.0 falls in that range.
A manager of a factory wants to know if a new quality check protocol has decreased the number of units a worker produces in a day. Before the new protocol, a worker could produce 27 units per day. What alternative hypothesis should the manager use to test this claim?
µ < 27 units is correct The manager wants to know if the new quality check protocol has decreased the average number of units a worker can produce per day. For a one-sided test, the manager should use the alternative hypothesis Ha: μ<27 units. This is the claim the manger wishes to substantiate.
Suppose the average satisfaction rating of the sample is 6.8 out of 10. Which of the following do you think would be the correct conclusion? Remember that H0:μ=6.7 and Ha:μ≠6.7
Do not reject the null hypothesis is correct If the average customer satisfaction rating has not changed (μ=6.7)(μ=6.7), it would not be unusual to draw a sample that has a mean of 6.8. Therefore, we would probably fail to reject the null hypothesis.
If you are performing a hypothesis test based on a 90% confidence level, what are your chances of making a type II error?
It is not possible to tell without more information
If the significance level of a hypothesis test is 10%, for which of the following p-values would you reject the null hypothesis? Select all that apply.
0.08, 0.05 is correct We reject the null hypothesis if the mean of our sample falls within the rejection region. The area of the rejection region is equal to the significance level, so we reject the null hypothesis when the p-value is less than the significance level. Since 0.08 is less than 0.10, we would reject the null hypothesis. Remember: the lower the p-value, the stronger the evidence is against the null hypothesis. Note that another option is also correct.
Now suppose we take a sample of 25 students, taking the same standardized test, which has a mean score of 500 and a standard deviation of 100, and find that the average score of this sample is 530. Which function would correctly calculate the 95% range of likely sample means under the null hypothesis?
500 ± CONFIDENCE.T(0.05,100,25) is correct The range of likely sample means is centered at the historical population mean, 500. Because our sample is less than 30, we cannot assume that the sample means are normally distributed, and so we should use CONFIDENCE.T rather than the CONFIDENCE.NORM function.
The owner of a local health food store recently started a new ad campaign to attract more business and wants to test whether average daily sales have increased. Historically average daily sales were approximately $2,700. After the ad campaign, the owner took another random sample of forty-five days and found that average daily sales were $2,984 with a standard deviation of approximately $585. Calculate the upper bound of the 95% range of likely sample means for this one-sided hypothesis test using the CONFIDENCE.NORM function.
=2700+CONFIDENCE.NORM(0.1,585,45)
Recall that the owner of a local health food store recently started a new ad campaign to attract more business and wants to know if average daily sales have increased. Historically average daily sales were approximately $2,700. The upper bound of the 95% range of likely sample means for this one-sided test is approximately $2,843.44. If the owner took a random sample of forty-five days and found that daily average sales were now $2,984, what can she conclude at the 95% confidence level?
Average daily sales have increased is correct Since the sample mean, $2,984, falls outside the range of likely sample means (which has an upper bound=$2,843.44), the store owner can reject the null hypothesis that μ≤$2,700μ at a 95% confidence level. Since she can reject the null hypothesis, she can essentially accept the alternative hypothesis and conclude the average daily sales have increased.
A streaming music site changed its format to focus on previously unreleased music from rising artists. The site manager now wants to determine whether the number of unique listeners per day has changed. Before the change in format, the site averaged 131,520 unique listeners per day. Now, beginning three months after the format change, the site manager takes a random sample of 30 days and finds that the site has an average of 124,247 unique listeners per day. SELECT THE TWO ANSWERS below that represent the correct null and alternative hypotheses.
Null hypotheses: μ=131,520 Alternative hypotheses: μ≠131,520
A streaming music site changed its format to focus on previously unreleased music from rising artists. The site manager now wants to determine whether the number of unique listeners per day has changed. Before the change in format, the site averaged 131,520 unique listeners per day. Now, beginning three months after the format change, the site manager takes a random sample of 30 days and finds that the site has an average of 124,247 unique listeners per day. The manager finds that the p-value for the hypothesis test is approximately 0.0743. How would you interpret the p-value?
If the average number of unique daily listeners per day is still 131,520, the likelihood of obtaining a sample with a mean at least as extreme as 124,247 is 7.43%. is correct The null hypothesis is that the average number of unique daily listeners per day has not changed, that is, it is still 131,520. Therefore, the p-value of 0.0743 indicates that if the average number of unique daily listeners is still 131,520, the likelihood of obtaining a sample with a mean at least as extreme as 124,247 is 7.43%%.
Suppose we want to know whether students who attend a top business school have higher earnings than those who attend lower-ranked business schools. To find out, we collect the average starting salaries of recent graduates from the top 100 business schools in the U.S. We then compare the salaries of those who attended the schools ranked in the top 50 to the salaries of those who did not. Should we perform a one-sided hypothesis test or a two-sided test?
One-sided is correct Since we are interested only in whether the average salaries of people who attended the top 50 business schools are higher than the salaries of those who did not, we should perform a one-sided test. If we were interested in learning whether the salaries of the people who went to the top 50 business schools were different (either higher or lower) than those from the other schools, we would conduct a two-sided test.
Suppose the average satisfaction rating of the sample is 9.9 out of 10. Which of the following do you think would be the correct conclusion? Remember that H0:μ=6.7and Ha:μ≠6.7
Reject the null hypothesis is correct The null hypothesis is that the average satisfaction rating has not changed, that is, that the population mean μ is still equal to 6.7. Drawing a sample with an average satisfaction rating of 9.9 from a population that has an average rating of 6.7 is extremely unlikely, so we would almost certainly reject the null hypothesis and conclude that the average satisfaction rating is no longer 6.7.
Suppose again that the movie theater manager had gathered a sample that had an average customer satisfaction rating of 7.05 but in this case had firm convictions that if the average rating had changed, it had increased. Given what you know about the relationship between the p-values of one-sided and two-sided tests, would you reject or fail to reject the null hypothesis, H0:μ≤6.7, at a 5% significance level? As noted above, for a two-sided test with H0:μ=6.7 and Ha:μ≠6.7, the p-value of 7.05 is approximately 0.07.
Reject the null hypothesis is correct The p-value for a one-sided hypothesis test is half the p-value of a two-sided test for the same value. The p-value for 7.05 for the two-sided hypothesis test was 0.07, so the p-value for 7.05 for the one-sided test is 0.035. Because 0.035 is less than the significance level, 0.05, we reject the null hypothesis and conclude that the average customer satisfaction rating has increased. Note that the outcomes of one-sided and two-sided tests can be different. Just because we did not reject the null hypothesis for the two-sided test does not mean that we will have the same result for the one-sided test.
We have found that for the movie theater example, the p-value for the one-sided hypothesis test is 0.0013. Assuming a 0.05 significance level, what would you conclude?
Reject the null hypothesis and conclude that the average satisfaction rating has increased is correct Because the p-value is less than the specified significance level of 0.05, we reject the null hypothesis. Our alternative hypothesis, the claim we wish to substantiate, is μ>6.7, so by rejecting the null hypothesis we are able to conclude that the average satisfaction rating has increased.
A streaming music site changed its format to focus on previously unreleased music from rising artists. The site manager now wants to determine whether the number of unique listeners per day has changed. Before the change in format, the site averaged 131,520 unique listeners per day. Now, beginning three months after the format change, the site manager takes a random sample of 30 days and finds that the site now has an average of 124,247 unique listeners per day. The manager finds that the p-value for the hypothesis test is approximately 0.0743. What can be concluded at the 95% confidence level?
The manager should fail to reject the null hypothesis; there is not enough evidence to conclude that the number of unique daily listeners has changed. is correct Since the p-value, 0.0743, is greater than the significance level, 0.05, the manager should fail to reject the null hypothesis.
A college student is interested in testing whether business majors or liberal arts majors are better at trivia. The student gives a trivia quiz to a random sample of 30 business school majors and finds the sample's average test score is 86. He gives the same quiz to 30 randomly selected liberal arts majors and finds the sample's average quiz score is 89. The student finds that the p-value for the hypothesis test equals approximately 0.0524. What can be concluded at αα=5%?
The student should fail to reject the null hypothesis and conclude that there is insufficient evidence of difference between business and liberal arts majors' knowledge of trivia. is correct Since the p-value, 0.0524, is greater than the significance level, 0.05, the student should fail to reject the null hypothesis and conclude that there is insufficient evidence of difference between business and liberal arts majors' knowledge of trivia. Because the null hypothesis is that there is no difference between the two types of majors, this answer is correct.
A food truck operator has traditionally sold 75 bowls of noodle soup each day. He moves to a new location and after a week sees that he has averaged 85 bowls of noodle soup sales each day. He runs a one-sided hypothesis test to determine if his daily sales at the new location have increased. The p-value of the test is 0.031. How should he interpret the p-value?
There is a 3.1% chance of obtaining a sample with a mean of 85 or higher assuming that the true mean sales at the new location is still equal to or less than 75 bowls a day. is correct The p-value provides us with the likelihood of the sample value equal to or more extreme than the observed sample value if the null hypothesis is true. In this case the p-value of 0.031 tells us that there would be a 3.1% chance of the sample value of 85 or above being observed if the null hypothesis were true.
An engineer designing a new type of bridge wants to test the stress and load bearing capabilities of a prototype before beginning construction. Her null hypothesis is that the bridge's stress and load capabilities are safe. Select which type of error would be worse. Make sure that the type of error is matched with the correct definition.
Type II; the engineer deems the bridge safe and moves onto construction even though it is not actually safe is correct The type II error is that the engineer deems the bridge safe and moves onto construction even though it is not actually safe. This would be worse than presuming that a safe bridge is unsafe.x
A manager of a factory wants to know if the average number of workplace accidents is different for workers who attended an equipment safety training compared to those who did not attend. What null hypothesis should the manager use to test this claim?
µattended = µdid not attend is correct If the manager's alternative hypothesis is that the average number of workplace accidents has changed between the two groups of workers, then the null hypothesis is that the average number of accidents has remained the same.
A manager of a factory wants to know if the average number of workplace accidents is different for workers who attended an equipment safety training compared to those who did not attend. What alternative hypothesis should the manager use to test this claim?
µattended ≠ µdid not attend is correct The manager has reason to believe that the training has changed the average number of workplace accidents between the two groups of workers. For a two-sided test, the manager should use the alternative hypothesis Ha: µattended ≠ µdid not attend. This is the claim the manger wishes to substantiate.
The manager now has reason to believe that showing old classics has increased the customer satisfaction rating. For this one-sided hypothesis test, what alternative hypothesis should he use?
μ>6.7 is correct The manager has reason to believe that the new artistic approach has increased the average customer satisfaction, so for a one-sided test he should use the alternative hypothesis Ha:μ>6.7. This is the claim he wishes to substantiate.
A college student is interested in testing whether business majors or liberal arts majors are better at trivia. The student gives a trivia quiz to a random sample of 30 business majors and finds the sample's average score is 86. He gives the same quiz to 30 randomly selected liberal arts majors and finds the sample's average score is 89. What is the alternative hypothesis of this test?
μBusiness≠μLiberal ArtsμBusiness≠μLiberal Artsis correct The alternative hypothesis is the claim that is being tested. Since the student wants to test whether there is a difference between business school majors' and liberal arts majors' trivia scores, the alternative hypothesis is that the mean scores are not equal.
Suppose we want to know whether students who attend a top business school have higher earnings. What is the alternative hypothesis?
μtop 50>μnot top 50 is correct The alternative hypothesis is the claim we wish to substantiate. In this case, we want to establish that people who attended a school ranked in the top 50 earn more than those who did not, so μtop 50>μnot top 50.
Suppose we want to know whether students who attend a top business school have higher earnings. What is the null hypothesis?
μtop 50≤μnot top 50 is correct The null hypothesis is the claim we assume to be true. It is the opposite of the alternative hypothesis—the claim we wish to substantiate. In this case, our alternative hypothesis is that people who attended a school ranked in the top 50 earn more than those who did not. The opposite of this is that people who attended a school ranked in the top 50 earn less than or equal to those who did not.
The owner of a local health food store recently started a new ad campaign to attract more business and wants to test whether average daily sales have increased. Historically average daily sales were approximately $2,700. After the ad campaign, the owner took a random sample of forty-five days and found that daily average sales had increased to $2,984. What is store owner's null hypothesis?
μ≤$2,700is correct The null hypothesis is the opposite of the hypothesis you are trying to substantiate. Since the owner wants to test for an increase, the null hypothesis is μ≤$2,700. Remember that the null hypothesis is always based on historical information.
For the one-sided hypothesis test, what should the movie theater manager use as the null hypothesis?
μ≤6.7 is correct If our alternative hypothesis is that the average satisfaction rating has increased, then the null hypothesis is that the rating is the same or lower. Thus, if our alternative hypothesis is that μ>6.7, our null hypothesis is that μ≤6.7