Quizlet when the p-value is used for hypothesis testing, the null hypothesis is rejected if

0.08
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.
0.89
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 0.89 is greater than 0.10, we would fail to reject the null hypothesis.
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 0.05 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.
0.11
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 0.11, is greater than 0.10, we would fail to reject the null hypothesis.

T TEST

Previously, based on the sample standard deviation of 2.8, the sample size of 196, and the confidence level of 95%, we found that the range of likely sample means runs from about 6.3 to about 7.1. Since our sample mean of 7.3 fell outside of that range, we concluded that we had sufficient evidence to reject the null hypothesis that the mean customer service rating had not changed.

Now let's calculate the p-value for our sample of movie theater customers and find out exactly how unlikely it would be to select a sample that has an average customer satisfaction rating at least as extreme as 7.3, if the average customer satisfaction rating is actually still 6.7. "At least as extreme" means at least as far from 6.7 as 7.3 is, that is, outside of the range 6.7±0.6. Thus, in this case, the likelihood that we would obtain a sample at least as extreme as 7.3 is the likelihood of obtaining a sample less than or equal to 6.1 or greater than or equal to 7.3.

Although there are multiple ways to calculate a p-value in Excel, we will use a t-test, the most common method used for hypothesis tests. The t-test uses a t-distribution, which provides a more conservative estimate of the p-value when the sample size is small. Recall that as the sample size increases, the t-distribution converges to a normal distribution, so a t-distribution can be used for large samples as well. Companies tend to use the t-distribution rather than the normal distribution because it is safe for both small and large samples.

Unfortunately, the process for conducting a one-sample hypothesis test in Excel is a bit unwieldy. To use Excel's T.TEST function for a hypothesis test with one sample, we must create a second column of data that will act as a second sample. We will walk through how to use the T.TEST function but please understand that the most important thing to take away from this discussion is the interpretation of the p-value.

=T.TEST(array1, array2, tails, type)

array1 is a set of numerical values or cell references. We will place our sample data in this range.
array2 is a set of numerical values or cell references. We have only one set of data, so we will use the historical mean, 6.7, as the second data set. To do this, we create a column with each entry equal to 6.7.
tails is the number of tails for the distribution. It can be either 1 or 2. We will learn more about what this means later in the module. Since our alternative hypothesis is that the mean has changed and therefore can be either lower or higher than the historical mean, we will be using a two-tailed, or two-sided hypothesis test.
type can be 1, 2, or 3. Type 1 is a paired test and is used when the same group is tested twice to provide paired "before and after" data for each member of the group. Type 2 is an unpaired test in which the samples are assumed to have equal variances. Type 3 is an unpaired test in which the samples are assumed to have unequal variances. The variances of the two columns are clearly different in our case, so we use type 3. There are ways to test whether variances are equal, but when in doubt, use type 3.

T TEST
Previously, based on the sample standard deviation of 2.8, the sample size of 196, and the confidence level of 95%, we found that the range of likely sample means runs from about 6.3 to about 7.1. Since our sample mean of 7.3 fell outside of that range, we concluded that we had sufficient evidence to reject the null hypothesis that the mean customer service rating had not changed.

Now let's calculate the p-value for our sample of movie theater customers and find out exactly how unlikely it would be to select a sample that has an average customer satisfaction rating at least as extreme as 7.3, if the average customer satisfaction rating is actually still 6.7. "At least as extreme" means at least as far from 6.7 as 7.3 is, that is, outside of the range 6.7±0.6. Thus, in this case, the likelihood that we would obtain a sample at least as extreme as 7.3 is the likelihood of obtaining a sample less than or equal to 6.1 or greater than or equal to 7.3.

Although there are multiple ways to calculate a p-value in Excel, we will use a t-test, the most common method used for hypothesis tests. The t-test uses a t-distribution, which provides a more conservative estimate of the p-value when the sample size is small. Recall that as the sample size increases, the t-distribution converges to a normal distribution, so a t-distribution can be used for large samples as well. Companies tend to use the t-distribution rather than the normal distribution because it is safe for both small and large samples.

Unfortunately, the process for conducting a one-sample hypothesis test in Excel is a bit unwieldy. To use Excel's T.TEST function for a hypothesis test with one sample, we must create a second column of data that will act as a second sample. We will walk through how to use the T.TEST function but please understand that the most important thing to take away from this discussion is the interpretation of the p-value.

=T.TEST(array1, array2, tails, type)

array1 is a set of numerical values or cell references. We will place our sample data in this range.
array2 is a set of numerical values or cell references. We have only one set of data, so we will use the historical mean, 6.7, as the second data set. To do this, we create a column with each entry equal to 6.7.
tails is the number of tails for the distribution. It can be either 1 or 2. We will learn more about what this means later in the module. Since our alternative hypothesis is that the mean has changed and therefore can be either lower or higher than the historical mean, we will be using a two-tailed, or two-sided hypothesis test.
type can be 1, 2, or 3. Type 1 is a paired test and is used when the same group is tested twice to provide paired "before and after" data for each member of the group. Type 2 is an unpaired test in which the samples are assumed to have equal variances. Type 3 is an unpaired test in which the samples are assumed to have unequal variances. The variances of the two columns are clearly different in our case, so we use type 3. There are ways to test whether variances are equal, but when in doubt, use type 3.

Fail to reject the null hypothesis
Because the p-value is less than the specified significance level of 0.05, we reject the null hypothesis. What can we conclude after rejecting it?
Reject the null hypothesis and conclude that the average satisfaction rating is no longer 6.7
Because the p-value is less than the specified significance level of 0.05, we reject the null hypothesis. If this were a two-sided hypothesis test, the accurate conclusion would be that the average satisfaction rating is no longer equal to 6.7. For this one-sided test, what is the alternative hypothesis, the claim we wish to substantiate?
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.
Reject the null hypothesis and conclude that the average satisfaction rating has decreased
Because the p-value is less than the specified significance level of 0.05, we reject the null hypothesis. For this one-sided test, what is the alternative hypothesis, the claim we wish to substantiate?

=T.TEST(array1, array2, tails, type)

array1 is a set of numerical values or cell references. We will place our sample data in this range.
array2 is a set of numerical values or cell references. We have only one set of data, so we will use the historical mean, 6.7, as the second data set. To do this, we create a column with each entry equal to 6.7.
tails is the number of tails for the distribution. It can be either 1 or 2. Now that we are performing a one-sided test, we will enter a 1 instead of a 2.
type can be 1, 2, or 3. Type 1 is a paired test and is used when the same group is tested twice to provide paired "before and after" data for each member of the group. Type 2 is an unpaired test in which the samples are assumed to have equal variances. Type 3 is an unpaired test in which the samples are assumed to have unequal variances. The variances of the two columns are clearly different in our case, so we use type 3. There are ways to test whether variances are equal, but when in doubt, use type 3.

The p-value of our sample, which has a mean of 7.3, is T.TEST(A2:A197,B2:B197,1,3)=0.0013.

Notice that 0.0013, the p-value from our one-sided test for the sample mean 7.3, is half of 0.0026, the p-value from our two-sided test for the sample mean 7.3. This should make sense. In each case, the p-value is the probability of obtaining a sample mean at least as extreme as 7.3 under the assumption that the null hypothesis is true. In the two-sided test, this is the probability of obtaining a sample with a mean less than 6.1 or greater than 7.3; in the one-sided test it is the probability of obtaining a sample with a mean greater than 7.3.

Thus, we can perform a two-sided hypothesis test and just divide the resulting p-value by two to obtain the p-value for the one-sided test.

530 ± CONFIDENCE.NORM(0.05,100,25)
The range of likely sample means is centered at the historical population mean, 500, not at the sample mean, 530. In addition, because of the small sample size, we cannot assume that the sample means are normally distributed, so we should not use the CONFIDENCE.NORM function.
530± CONFIDENCE.T(0.05,100,25)
The range of likely sample means is centered at the historical population mean, 500, not at the sample mean, 530.
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.
500 ± CONFIDENCE.NORM(0.05,100,25)
Because of the small sample size, we cannot assume that the sample means are normally distributed, so we should not use the CONFIDENCE.NORM function.
Result

Correct!

Question 2 of 11

3.2 Summary

Lesson Summary

We use hypothesis tests to substantiate a claim about a population mean (or other population parameter).

The null hypothesis (H0) is a statement about a topic of interest about the population. 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 using evidence from a sample. The null hypothesis is the opposite of the hypothesis we are trying to prove (the alternative hypothesis).
The alternative hypothesis (Ha) is the theory or claim we are trying to substantiate.

Before conducting a hypothesis test:

Determine whether to analyze a change in a single population or compare two populations.
We perform a single-population hypothesis test when we want to determine whether a population's mean is significantly different from its historical average.
We perform a two-population hypothesis test when we want to compare the means of two populations—for example, when we want to conduct an experiment and test for a difference between a control and treatment group.
Determine whether to perform a one-sided or two-sided hypothesis test.
We perform two-sided tests when we do not have strong convictions about the direction of a change. Therefore we test for a change in either direction
We perform a one-sided test when we have strong convictions about the direction of a change—that is, we know that the change is either an increase or a decrease.

To conduct a hypothesis test, we must follow these steps:

State the null and alternative hypotheses.
Choose the level of significance for the test.
Gather data about a sample or samples.
To determine whether the sample is highly unlikely under the assumption that the null hypothesis is true, construct the range of likely sample means or calculate the p-value.
The p-value is the likelihood of obtaining a sample as extreme as the one we've obtained, if the null hypothesis is true.
The p-value of a one-sided hypothesis test is half the p-value of a two-sided hypothesis test.
If the sample mean falls in the range of likely sample means, or if its p-value is greater than the stated significance level, we do not have sufficient evidence to reject the null hypothesis.
If the sample mean falls in the rejection region, or if it has a p-value lower than the stated significance level, we have sufficient evidence to reject the null hypothesis. We can never accept the null hypothesis.
Trade-offs: The higher the confidence level (and therefore the lower the significance level), the lower the chance of rejecting the null hypothesis when it is true (type I error or false positive). But the higher the confidence level, the higher the chance of not rejecting it when it is false (type II error or false negative).

Excel Summary

Calculating the range of likely sample means using CONFIDENCE.NORM or CONFIDENCE.T
=T.TEST(array1, array2, tails, type)

3.2 Summary

Lesson Summary

We use hypothesis tests to substantiate a claim about a population mean (or other population parameter).

The null hypothesis (H0) is a statement about a topic of interest about the population. 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 using evidence from a sample. The null hypothesis is the opposite of the hypothesis we are trying to prove (the alternative hypothesis).
The alternative hypothesis (Ha) is the theory or claim we are trying to substantiate.

Before conducting a hypothesis test:

Determine whether to analyze a change in a single population or compare two populations.
We perform a single-population hypothesis test when we want to determine whether a population's mean is significantly different from its historical average.
We perform a two-population hypothesis test when we want to compare the means of two populations—for example, when we want to conduct an experiment and test for a difference between a control and treatment group.
Determine whether to perform a one-sided or two-sided hypothesis test.
We perform two-sided tests when we do not have strong convictions about the direction of a change. Therefore we test for a change in either direction
We perform a one-sided test when we have strong convictions about the direction of a change—that is, we know that the change is either an increase or a decrease.

To conduct a hypothesis test, we must follow these steps:

State the null and alternative hypotheses.
Choose the level of significance for the test.
Gather data about a sample or samples.
To determine whether the sample is highly unlikely under the assumption that the null hypothesis is true, construct the range of likely sample means or calculate the p-value.
The p-value is the likelihood of obtaining a sample as extreme as the one we've obtained, if the null hypothesis is true.
The p-value of a one-sided hypothesis test is half the p-value of a two-sided hypothesis test.
If the sample mean falls in the range of likely sample means, or if its p-value is greater than the stated significance level, we do not have sufficient evidence to reject the null hypothesis.
If the sample mean falls in the rejection region, or if it has a p-value lower than the stated significance level, we have sufficient evidence to reject the null hypothesis. We can never accept the null hypothesis.
Trade-offs: The higher the confidence level (and therefore the lower the significance level), the lower the chance of rejecting the null hypothesis when it is true (type I error or false positive). But the higher the confidence level, the higher the chance of not rejecting it when it is false (type II error or false negative).

Excel Summary

Calculating the range of likely sample means using CONFIDENCE.NORM or CONFIDENCE.T
=T.TEST(array1, array2, tails, type)

550 ± CONFIDENCE.NORM(0.05,100,100)
The range of likely sample means is centered at the historical population mean, 500, not at the sample mean, 550.
550± CONFIDENCE.T(0.05,100,100)
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. In addition, the range of likely sample means is centered at the historical population mean, 500, not at the sample mean, 550.
500 ± CONFIDENCE.T(0.05,100,100)
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.
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.

530 ± CONFIDENCE.NORM(0.05,100,25)
The range of likely sample means is centered at the historical population mean, 500, not at the sample mean, 530. In addition, because of the small sample size, we cannot assume that the sample means are normally distributed, so we should not use the CONFIDENCE.NORM function.
530± CONFIDENCE.T(0.05,100,25)
The range of likely sample means is centered at the historical population mean, 500, not at the sample mean, 530.
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.
500 ± CONFIDENCE.NORM(0.05,100,25)
Because of the small sample size, we cannot assume that the sample means are normally distributed, so we should not use the CONFIDENCE.NORM function.

550 ± CONFIDENCE.NORM(0.05,100,100)
The range of likely sample means is centered at the historical population mean, 500, not at the sample mean, 550.
550± CONFIDENCE.T(0.05,100,100)
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. In addition, the range of likely sample means is centered at the historical population mean, 500, not at the sample mean, 550.
500 ± CONFIDENCE.T(0.05,100,100)
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.
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.
Result

When the p

Can the p

How do you reject a hypothesis with p

What does rejecting the p