BA Module 3
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 The one-sided p-value is half of the two-sided p-value. Thus, the one-sided p-value is 0.00202=0.00100.00202=0.0010.
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?
0.0020 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?
0.0300 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.
On the basis of the resulting p-value, would we reject the null hypothesis or fail to reject the null hypothesis at the 0.05 significance level?
Reject the null hypothesis Because the p-value, 0.0000, is less than the significance level, we should reject the null hypothesis.
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. 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.
t-test (p-value)
the most common method used for hypothesis tests.
Alternative Hypothesis (Ha):
The alternative hypothesis (the opposite of the null hypothesis) is the theory or claim we are trying to substantiate. If our data allow us to nullify the null hypothesis, we substantiate the alternative hypothesis.
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 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>6.7Ha:μ>6.7. This is the claim he wishes to substantiate.
If you are performing a hypothesis test based on a 20% significance level, what are your chances of making a type I error?
20% The probability of a type I error is equal to the significance level, which is 1-confidence level.
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=6.7H0:μ=6.7 and Ha:μ≠6.7Ha:μ≠6.7.
Do not reject the null hypothesis 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.
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 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.
observational study
researchers observe and collect data about a sample (e.g., people or items) as they occur naturally, without intervention, and analyze the data to investigate possible relationships.
Select the p-value(s) at which you would reject the null hypothesis for a two-sided test at the 90% confidence level. SELECT ALL THAT APPLY.
0.0250 To reject the null hypothesis at the 90% confidence level, the p-value must be less than 1-0.90=0.10. 0.0250 is less than 0.10 so we can reject the null hypothesis. 0.0500 To reject the null hypothesis at the 90% confidence level, the p-value must be less than 1-0.90=0.10. 0.0500 is less than 0.10 so we can reject the null hypothesis. 0.0900 To reject the null hypothesis at the 90% confidence level, the p-value must be less than 1-0.90=0.10. 0.0900 is less than 0.10 so we can 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 I error?
10%
What is the significance level for a 95% confidence level?
5% Significance level=1-confidence level. 1-0.95=0.05, that is, 5%.
Now suppose we take a sample and find the average satisfaction rating to be 7.3. What should be the center of the range of likely sample means? Remember that H0:μ=6.7H0:μ=6.7 and Ha:μ≠6.7Ha:μ≠6.7.
6.7 We always start a hypothesis test by assuming that the null hypothesis is true. Thus, the center of the range of likely sample means is the historical average—the average specified by the null hypothesis, in this case is 6.7. Remember, the null hypothesis is that showing old classics has not changed the average satisfaction rating.
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 thatμ=6.7 and Ha:μ≠6.7Ha:μ≠6.7.
Do not reject the null hypothesis 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.
When performing a hypothesis test based on a 95% confidence level, what are the chances of making a type II error?
It is not possible to tell without more information. A type II error occurs when we fail to reject the null hypothesis when the null hypothesis is actually false. The confidence level does not provide any information about the likelihood of making a type II error. Calculating the chances of making a type II error is quite complex and beyond the scope of this course.
One-sided hypothesis test vs Two-sided hypothesis test
One-sided hypothesis test TEST WHETHER INCOMING STUDENTS AT A BUSINESS SCHOOL RECEIVE BETTER GRADES IN THEIR CLASSES IF THEY'VE TAKEN AN ON-LINE PROGRAM COVERING BASIC MATERIAL TEST WHETHER USERS OF A COMMERCIAL WEBSITE ARE LESS LIKELY TO MAKE A PURCHASE IF THEY ARE REQUIRED TO SET UP A USER ACCOUNT ON THE SITE Two-sided hypothesis test TEST WHETHER THERE IS A DIFFERENCE BETWEEN MEN'S AND WOMEN'S USAGE OF A MOBILE FITNESS APP TEST WHETHER THE NUMBER OF LISTENERS OF A STREAMING MUSIC SERVICE HAS CHANGED AFTER THEY CHANGED THE USER INTERF
Suppose the average satisfaction rating of the sample is 3.5 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.
Reject the null hypothesis The null hypothesis is that the average satisfaction rating has not changed (μ=6.7)(μ=6.7). Drawing a sample with an average satisfaction rating of 3.5 from a population that has an average rating of 6.7 is extremely unlikely, so we would reject the null hypothesis and conclude that the average satisfaction rating is no longer 6.7. Note that 3.5 is the same distance (3.2) from 6.7 as 9.9 is from 6.7. Since the distribution of sample means is symmetric, we can conclude that 3.5 and 9.9 have the same (very low) likelihood of being drawn from a population with a mean of 6.7. We will see shortly the key roles the distribution of sample means and the central limit theorem play in hypothesis testing.
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 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>6.7μ>6.7, so by rejecting the null hypothesis we are able to conclude that the average satisfaction rating has increased.
A business school professor is interested to know if watching a video about the Central Limit Theorem helps students understand it. To assess this, the professor tests students' knowledge both immediately before they watch the video and immediately after. The professor takes a sample of students, and for each one compares their test score after the video to their score before the video. Using the data below, calculate the p-value for the following hypothesis test: H0:μafter≤μbefore Ha:μafter>μbefore
The p-value of the one-sided hypothesis test is T.TEST(array1, array2, tails, type)=T.TEST(B2:B31,C2:C31,1,1), which is approximately 0.0128. You must designate this test as a one-sided test (that is, assign the value 1 to the tails argument) and as a type 1 (a paired test) because you are testing the same students on the same knowledge at two points in time. You must link directly to values in order to obtain the correct answer.
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 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.
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 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.
Before we determine the significance of the results, let's look at the direction of the change. What is the effect of changing the shopping cart design on Total Units Ordered (Units) and Ordered Product Sales (OPS)?
Units increased and OPS increased For each test, the mean of the treatment is larger than the mean of the control. This indicates that changing the design of the shopping cart increased both Units and OPS. The Mean Difference and % Mean Difference confirm the increases.
Are the results for Total Units Ordered (Units) significant at the 95% confidence level?
Yes A test's results are significant if the p-value is less than the significance level. In this case, the p-value, 0.0169, is less than 0.05, so the results are significant.
Are the results for Ordered Product Sales (OPS) significant at the 95% confidence level?
Yes A test's results are significant if the p-value is less than the significance level. In this case, the p-value, 0.0339, is less than 0.05, so the results are significant.
type II error
false negative (we incorrectly fail to reject the null hypothesis when it is actually not tru
type I error
often called a false positive (we incorrectly reject the null hypothesis when it is actually true)
survey
researchers ask questions and record self-reported responses from a random sample of a population.
experiment
researchers divide a sample into two or more groups. One group is a "control group," which has not been manipulated. In the "treatment group (or groups)," they manipulate a variable and then compare the treatment group(s) responses to the responses of the control group.
What is the alternative hypothesis (HaHa) of the movie theater example? Recall that the historical average customer satisfaction rating is 6.7 out of 10.
μ≠6.7 The alternative hypothesis is that the new artistic approach of showing old classics has changed the average satisfaction rating. Therefore Ha:μ≠6.7Ha:μ≠6.7. Note that Ha:μ≠6.7Ha:μ≠6.7 is the opposite of H0:μ=6.7H0:μ=6.7, which confirms our understanding that the alternative hypothesis is the opposite of the null hypothesis
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) 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.
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 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.
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 This is the null hypothesis. Remember that the null and alternative hypotheses are opposites.
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) 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.
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 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≤$2,700 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. 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%. 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%%.
Since the p-value, 0.0026, is less than the 0.05 significance level, we reject the null hypothesis and conclude that the customer satisfaction rating has changed. 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% 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%.
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≤6.7H0:μ≤6.7, at a 5% significance level? As noted above, for a two-sided test with H0:μ=6.7H0:μ=6.7 and Ha:μ≠6.7Ha:μ≠6.7, the p-value of 7.05 is approximately 0.07.
Reject the null hypothesis 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.
Null Hypothesis (H0)
The null hypothesis is a statement about a topic of interest. It is typically based on historical information or conventional wisdom. We always start a hypothesis test by assuming that the null hypothesis is true and then test to see if we can nullify it—that's why it's called the "null" hypothesis. The null hypothesis is the opposite of the hypothesis we are trying to prove (the alternative hypothesis).
significance level
The significance level defines the rejection region by specifying the threshold for deciding whether or not to reject null hypothesis. When the p-value of a sample mean is less than the significance level, we reject the null hypothesis. The significance level is the area of the rejection region, meaning the area under the distribution of sample means over the rejection region. The significance level is the probability of rejecting the null hypothesis when the null hypothesis is actually true.
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. 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.
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.
To construct a 95% range of likely sample means, calculate the margin of error using the function CONFIDENCE.NORM(alpha, standard_dev, size). However, CONFIDENCE.NORM finds the margin of error for a two-sided hypothesis test and this question asks for the upper bound of a one-sided test. To find the upper bound for the one-sided test you must first determine what two-sided test would have a 5% rejection region on the right side. Since the distribution of sample means is symmetric, a two-sided test with a 10% significance level would have a 5% rejection region on the left side of the normal distribution and a 5% rejection region on the right side. Thus, the upper bound for a two-sided test with alpha=0.1 will be the same as the upper bound on a one-sided test with alpha=0.05. The margin of error is CONFIDENCE.NORM(alpha, standard_dev, size)= CONFIDENCE.NORM(0.1,C3,C4)=CONFIDENCE.NORM(0.1,585,45)=$143.44. The upper bound of the 95% range of likely sample means for this one-sided hypothesis test is the population mean plus the margin of error, which is approximately $2,700+$143.44=$2,843.44.
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=6.7H0:μ=6.7 and Ha:μ≠6.7Ha:μ≠6.7, the p-value is approximately 0.07. Would you reject or fail to reject the null hypothesis, μ=6.7μ=6.7, at the 5% significance level?
ail to reject the null hypothesis 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 _____________ hypothesis test to _____________.
one-sided, 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.
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) 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.
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% 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.
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) 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.
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) 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.
If we specify a 75% confidence level, what percentage of sample means do we expect to fall in the rejection region?
If we specify a 75% confidence level, what percentage of sample means do we expect to fall in the rejection region?
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
Let's return to the movie theater example and focus on the sample taken after the manager changes the theater's artistic focus. 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=6.7H0:μ=6.7 and Ha:μ≠6.7Ha:μ≠6.7.
Reject the null hypothesis The null hypothesis is that the average satisfaction rating has not changed, that is, that the population mean μμ 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.
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. 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 score is 86. He gives the same quiz to 30 randomly selected liberal arts majors and finds the sample's average score is 89. Using the data provided below, calculate the p-value for the following hypothesis test: H0:μBusiness=μLiberal ArtsH0:μBusiness=μLiberal Arts Ha:μBusiness≠μLiberal ArtsHa:μBusiness≠μLiberal Arts
The p-value of the two-sided hypothesis test is T.TEST(array1, array2, tails, type)=T.TEST(A2:A31,B2:B31,2,3), which is approximately 0.0524. You must designate this test as a two-sided test (that is, assign the value 2 to the tails argument) and as a type 3 test (an unpaired test with unequal variances) because you are testing two different samples. You must link directly to values in order to obtain the correct answer
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. 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.
The manager now has reason to believe that showing old classics has increased the customer satisfaction rating. Recall that the historical average satisfaction rating was 6.7 and that the random sample of 196 moviegoers has an average satisfaction rating of 7.3 and a standard deviation of 2.8. Calculate the upper bound of the 95% range of likely sample means for this one-sided hypothesis test using the CONFIDENCE.NORM function.
To find the upper bound for the one-sided test we must first determine what two-sided test would have a 5% rejection region on the right side. Since the distribution of sample means is symmetric, a two-sided test with a 10% significance level would have a 5% rejection region on the left side of the normal distribution and a 5% rejection region on the right side. Thus, the upper bound for a two-sided test with alpha=0.1 will be the same as the upper bound on a one-sided test with alpha=0.05. The margin of error is *CONFIDENCE.NORM(0.1,C3,C4)=0.33*. The upper bound of the 95% range of likely sample means for this one-sided hypothesis test is the population mean plus the margin of error, which is approximately 6.7+0.33=7.03.
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. Using the data provided below, calculate the p-value for the following hypothesis test: H0:μ=131,520H0:μ=131,520 Ha:μ≠131,520Ha:μ≠131,520
To use Excel's T.TEST function for a hypothesis test with one sample, you must create a second column of data that will act as a second sample. So, first enter the historical average unique listeners into each cell in column B associated with a day in the sample; that is, enter 131,520 into cells B2 to B31. Then, the p-value of the two-sided hypothesis test is T.TEST(array1, array2, tails, type)=T.TEST(A2:A31,B2:B31,2,3), which is approximately 0.0743. You must link directly to values in order to obtain the correct answer.
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 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 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.
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.
μ=131,520μ=131,520 The null hypothesis is that number of unique listeners per day has not changed. Thus, μ=131,520μ=131,520 is the null hypothesis. μ≠131,520μ≠131,520 The alternative hypothesis is that the number of unique listener per day has changed. Thus, μ≠131,520μ≠131,520 is the alternative hypothesis.
What is the null hypothesis (H0H0) of the movie theater example? Recall that the historical average customer satisfaction rating is 6.7 out of 10.
μ=6.7 The null hypothesis is that the new artistic approach of showing old classics has not affected the average customer satisfaction rating; that is, the new average customer satisfaction rating is the same as its historical value of 6.7 out of 10. Therefore H0:μ=6.7H0:μ=6.7.
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 Arts 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 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 50μtop 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 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.
For the one-sided hypothesis test, what should the movie theater manager use as the null hypothesis?
μ≤6.7μ≤6.7 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>6.7μ>6.7, our null hypothesis is that μ≤6.7μ≤6.7.
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≥$2,984μ≥$2,984 The null hypothesis is the opposite of the hypothesis you are trying to substantiate. The owner wants to test for an increase. In addition, the null hypothesis is always based on historical information.
