CHAPTER 10

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CHAPTER 10

SOLUTIONS TO PROBLEMS

The z test and z distribution may be used for making one sample hypothesis tests involving a population mean under two conditions: (1) if the population standard deviation (s) is known, or (2) if the sample size is large enough (n » 50 or more) so that the sample standard deviation (s) can be used as an unbiased estimate of the population standard deviation. If either of these two conditions do not exist, i.e., if s is unknown or if the sample size is small, then hypothesis tests about one population mean must be conducted with the t test and t distribution.
In our first hypothesis test, the null and alternative hypotheses would be:
```
H₀: m = $2,333
H₁: m ¹ $2,333
```
The null hypothesis states that the population mean is equal to $2,333. If we think the true dollar amount lost is different than this, we state the non-directional or two-tail alternative, which simply presumes that the population mean is not equal to $2,333.

If we believed the dollar amount lost by burglary victims to be higher than $2,333, our null hypothesis would be the same, but we would assume the following about the alternative hypothesis:

H₁: m > $2,333

In this one-tail or directional alternative hypothesis we state our assumption that the true population mean is greater than $2,333.
To do our hypothesis test, first we state the null and alternative hypothesis. The null hypothesis is that the mean number of cheating episodes is 4.6. Our alternative hypothesis is that the true population mean is greater than 4.6. We have presumed a direction with respect to the alternative hypothesis so we have a one-tailed alternative hypothesis. The null and alternative hypotheses are:

H₀: m = 4.6

H₁: m > 4.6

Now we need to select our test statistic and the sampling distribution of that test statistic. Although we do not know the population standard deviation, our sample size is greater than 50 (n = 64), so we can use the z test and the z distribution. The third step in a hypothesis test is to select a level of significance and determine both the critical value and the critical region of the test statistic. The problem tells you to use an alpha level of .01. Since we are conducting a one-tail hypothesis test that states that the population mean is greater than some given value, our critical value and critical region lie in the right tail of the z distribution. Going to the z table, with a=.01, we can see that our critical value of z is equal to 2.33. The critical region is the area to the right of this z value (shown in the picture).

Any obtained value of z greater than or equal to 2.33 falls in the critical region, and would lead us to reject the null hypothesis. Our decision rule, therefore, is to reject the null hypothesis if our obtained value of z is 2.33 or greater: reject H₀ if z_obt ³ 2.33.

We are now ready to calculate our obtained z value. We have a sample mean of 6.3, a sample standard deviation of 1.9, and a sample size of 64. We put these numbers into our z test formula and find z_obt:

Our obtained value of z is 7.08. Since this is greater than the critical value of 2.33, and falls in the critical region, we will reject the null hypothesis that the population mean is equal to 4.6 times.
We begin the hypothesis test by stating the null and alternative hypothesis. The null hypothesis is that the number of times in the population that adolescents commit an act of vandalism is 3.5. The alternative hypothesis is that the true population mean is less than 3.5. Since we have stated a direction in our alternative hypothesis, we will conduct a one-tail hypothesis test. The null and alternative hypotheses are:

H₀: m = 3.5

H₁: m < 3.5

Now we need to select our test statistic and the sampling distribution of that test statistic. Although we do not know the population standard deviation, our sample size is greater than 50 (n = 59), so we can use the z test and the z distribution.

The third step in a hypothesis test is to select a level of significance and determine both the critical value and the critical region of the test statistic. The problem tells you to use an alpha level of .05. Since we are conducting a one-tail hypothesis test that states that the population mean is less than some given value, our critical value and critical region lie in the left tail of the z distribution. Going to the z table, with a=.05, we can see that our critical value of z is equal to -1.65. The critical region is the area to the left of this z value (shown in the picture). Any obtained value of z less than or equal to -1.65 falls in the critical region, and would lead us to reject the null hypothesis. Our decision rule, therefore, is to reject the null hypothesis if our obtained value of z is -1.65 or less: reject H₀ if z_obt £ -1.65.

We are now ready to calculate our obtained z value. We have a sample mean of 2.9, a sample standard deviation of .7, and a sample size of 59. We put these numbers into our z test formula and find z_obt:

Our obtained value of z is -6.67. Since this is less than the critical value of -1.65, and falls in the critical region, we will reject the null hypothesis that the population mean is equal to 3.5 acts of vandalism.
Again, let's go though each step of our hypothesis test. First, we state the null and alternative hypothesis. The null hypothesis is that the mean sentence length for the population of convicted assaulters is 25.9 months. Our alternative hypothesis is that the true population mean is different than 25.9. We have not presumed a direction with respect to the alternative hypothesis so we have a two-tailed alternative hypothesis. The null and alternative hypotheses are:

H₀: m = 25.9

H₁: m ¹ 25.9

Now we need to select our test statistic and the sampling distribution of that test statistic. Although we do not know the population standard deviation, our sample size is greater than 50 (n = 75), so we can use the z test and the z distribution.

The third step in a hypothesis test is to select a level of significance and determine both the critical value and the critical region of the test statistic. The problem tells you to use an alpha level of .01. Since we are conducting a two-tail hypothesis test that states that the population mean is different than some given value, our critical values and critical regions lie in both the right and left tails of the z distribution. Since we split our alpha level in both tails of the distribution, µ/2 = .01/2 = .005 of the area of the curve is in each of the two critical regions. Going to the z table, with a=.01 for a two-tailed test, we can see that our critical values of z are equal to ±2.58. The critical regions are the area to the right of a z value of 2.58 and to the left of a z value of -2.58 (shown in the picture). Any obtained value of z either greater than or equal to 2.58 or less than or equal to -2.58 falls in the critical region, and would lead us to reject the null hypothesis. Our decision rule, therefore, is to reject the null hypothesis if our obtained value of z ³ 2.58 or if z obtained £ -2.58: reject H₀ if z_obt ³ 2.58 or z_obt £ -2.58.

We are now ready to calculate our obtained z value. We have a sample mean of 27.3, a sample standard deviation of 6.5, and a sample size of 75. We put these numbers into our z test formula and find z_obt:

Our obtained value of z is 1.87. Since this is not greater than the critical value of 2.58, and does not fall in the critical region, we fail to reject the null hypothesis that the population mean is equal to 25.9 months.
The first thing you have to do in order to solve this problem is to calculate the mean and standard deviation. This will give you some practice with those formulas that you first saw in Chapters 4 and 5. When you do this, you should have obtained a sample mean of 16.4 and standard deviation of 4.0. Now we can conduct our hypothesis test. We think it is a great idea to follow the steps of hypothesis testing step-by-step, so we will continue that process here.

The first step is to state the null and alternative hypothesis. The null hypothesis is that the average number of hours that inmates at state correctional facilities spend in their cells during a day is 11. Our alternative hypothesis is that the true population mean is different than 11 hours. We have not presumed a direction with respect to the alternative hypothesis so we have a two-tailed alternative hypothesis. The null and alternative hypotheses are:

H₀: m = 15

H₁: m ¹ 15

Now we need to select our test statistic and the sampling distribution of that test statistic. Since we do not know the population standard deviation, and our sample size is substantially less than 50 (n = 15), we must use the t test and the t distribution. The third step in a hypothesis test is to select a level of significance and determine both the critical value and the critical region of the test statistic. The problem tells you to use an alpha level of .05. Since we are entertaining a two-tail alternative hypothesis test that states that the population mean is different than some given value, our critical values and critical regions lie in both the right and left tails of the t distribution. In addition to an alpha level, the t distribution requires knowing the degrees of freedom. With an n of 15, we have 15 - 1 or 14 degrees of freedom. Going to the t table with 14 degrees of freedom, an alpha of .05, and a two-tailed test, we can see that the critical value of t is ±2.145. The critical regions are the area to the right of a t value of 2.145 and to the left of a t value of -2.145 (shown in the picture).

Any obtained value of t either greater than or equal to 2.145 or less than or equal to -2.145 falls in the critical region, and would lead us to reject the null hypothesis. Our decision rule, therefore, is to reject the null hypothesis if our obtained value of t ³ 2.145 or if t obtained £ -2.145: reject H₀ if t_obt ³ 2.145 or t_obt £ -2.145.

We are now ready to calculate our obtained t value. We have a sample mean of 16.4, a sample standard deviation of 4.0, and a sample size of 15. We put these numbers into our t test formula and find t_obt:

Our obtained value of t is 1.31. Since this is not greater than the critical value of 2.145, and does not fall in the critical region, we fail to reject the null hypothesis that the population mean is equal to 15 hours.
You are given the sample mean and standard deviation in this problem, and are asked to test the null hypothesis that m = 4. First, state the null and alternative hypothesis. The null hypothesis is that the average number of arrests made by police officers in the town is 4 per year. The alternative hypothesis is that the true population mean is greater than 4 arrests. We have presumed a direction with respect to the alternative hypothesis so we have a one-tailed alternative hypothesis. The null and alternative hypotheses are:

H₀: m = 4

H₁: m > 4

Now we need to select our test statistic and the sampling distribution of that test statistic. Since we do not know the population standard deviation, and our sample size is substantially less than 50 (n = 12), we must use the t test and the t distribution.

The third step in a hypothesis test is to select a level of significance and determine both the critical value and the critical region of the test statistic. The problem tells you to use an alpha level of .01. To use the t distribution you also need the appropriate degrees of freedom. With an n of 12, we have 12 - 1 or 11 degrees of freedom. Going to the t table with 11 degrees of freedom, an alpha of .01, and a one-tailed test, we can see that the critical value of t is 2.718. The critical region is the area to the right of a t value of 2.718 (shown in the picture below).

Any obtained value of t greater than or equal to 2.718 falls in the critical region, and would lead us to reject the null hypothesis. Our decision rule, therefore, is to reject the null hypothesis if our obtained value of t ³ 2.718: reject H₀ if t_obt ³ 2.718.

We are now ready to calculate our obtained t value. We have a sample mean of 6.3, a sample standard deviation of 1.5, and a sample size of 12. We put these numbers into our t test formula and find t_obt:

Our obtained value of t is 5.11. Since this is greater than the critical value of 2.718, and falls in the critical region, we decide to reject the null hypothesis that the population mean is equal to 4 arrests.
As in the last problem, you are given the sample mean and standard deviation, and are asked to test the null hypothesis that m = 25 minutes. As usual, we will go through each step of the hypothesis test. First, state the null and alternative hypothesis. The null hypothesis is that the average length of a juvenile court hearing is 25 minutes. The alternative hypothesis is that the true population mean is less than 25 minutes. We have presumed a direction with respect to the alternative hypothesis so we have a one-tailed alternative hypothesis. The null and alternative hypotheses are:

H₀: m = 25

H₁: m < 25

Now we need to select our test statistic and the sampling distribution of that test statistic. Since we do not know the population standard deviation, and our sample size is substantially less than 50 (n = 20), we must use the t test and the t distribution.

The third step in a hypothesis test is to select a level of significance and determine both the critical value and the critical region of the test statistic. The problem tells you to use an alpha level of .05. To use the t distribution you also need the appropriate degrees of freedom. With an n of 20, we have 20 - 1 or 19 degrees of freedom. Going to the t table with 19 degrees of freedom, an alpha of .05, and a one-tailed test, we can see that the critical value of t is -1.729. The critical region is the area to the left of a t value of -1.729 (shown in the picture below).

Any obtained value of t less than or equal to -1.729 falls in the critical region, and would lead us to reject the null hypothesis. Our decision rule, therefore, is to reject the null hypothesis if our obtained value of t £ -1.729: reject H₀ if t_obt £ -1.729.

We are now ready to calculate our obtained t value. We have a sample mean of 23, a sample standard deviation of 6, and a sample size of 20. We put these numbers into our t test formula and find t_obt:

Our obtained value of t is -1.45. Since this is not less than the critical value of -1.729, and does not fall in the critical region, we fail to reject the null hypothesis that the population mean is equal to 25 minutes.
We begin the hypothesis test by stating the null and alternative hypothesis. The null hypothesis is that the proportion of American homes with firearms is .45. The alternative hypothesis is that the true population proportion is less than .45. Since we have stated a direction in our alternative hypothesis, we will conduct a one-tail hypothesis test. The null and alternative hypotheses are:

H₀: p = .45

H₁: p < .45

Now we need to select our test statistic and the sampling distribution of that test statistic. Since this is a problem involving a population proportion with a large sample size (n=200), we can use the z test and the z distribution.

The third step in a hypothesis test is to select a level of significance and determine both the critical value and the critical region of the test statistic. The problem tells you to use an alpha level of .01. Since we are conducting a one-tail hypothesis test that states that the population proportion is less than some given value, our critical value and critical region lie in the left tail of the z distribution. Going to the z table, with a=.01, we can see that our critical value of z is equal to -2.33 The critical region is the area to the left of this z value (shown in the picture).

Any obtained value of z less than or equal to -2.33 falls in the critical region, and would lead us to reject the null hypothesis. Our decision rule, therefore, is to reject the null hypothesis if our obtained value of z is -2.33 or less: reject H₀ if z_obt £ -2.33.

We are now ready to calculate our obtained z value. We have a population proportion of .45, a sample proportion of .23, and a sample size of 200. We put these numbers into our z test formula and find z_obt:

Our obtained value of z is -6.29. Since this is less than the critical value of -2.33, and falls in the critical region, we will reject the null hypothesis that the population proportion is equal to .45 or 45%.
First, we state the null and alternative hypothesis. The null hypothesis is that the proportion of homes in the neighborhood that have been victimized is .20. Our alternative hypothesis is that the true population proportion is different than .20. We have not presumed a direction with respect to the alternative hypothesis so we have a two-tailed alternative hypothesis. The null and alternative hypotheses are:

H₀: p = .20

H₁: p ¹ .20

Now we need to select our test statistic and the sampling distribution of that test statistic. Since this is a problem involving a population proportion with a large sample size (n=60; 60 x .2 > 5), we can use the z test and the z distribution. The next step is to select a level of significance and determine both the critical value and the critical region of the test statistic. The problem tells you to use an alpha level of .05. Since we are conducting a two-tail hypothesis test that states that the population mean is different than some given value, our critical values and critical regions lie in both the right and left tails of the z distribution. Since we split our alpha level in both tails of the distribution, µ/2 = .05/2 = .025 of the area of the curve is in each of the two critical regions. Going to the z table, with a=.05 for a two-tailed test, we can see that our critical values of z are equal to ±1.965. The critical regions are the area to the right of a z value of 1.96 and to the left of a z value of -1.96 (shown in the picture).

Any obtained value of z either greater than or equal to 1.96 or less than or equal to -1.96 falls in the critical region, and would lead us to reject the null hypothesis. Our decision rule, therefore, is to reject the null hypothesis if our obtained value of z ³ 1.96 or if z obtained £ -1.96: reject H₀ if z_obt ³ 1.96 or z_obt £ -1.96.

We are now ready to calculate our obtained z value. We have a population proportion of .20, a sample proportion of .31, and a sample size of 60. We put these numbers into our z test formula and find z_obt:

Our obtained value of z is 2.20. Since this is greater than the critical value of 1.96, and falls in the critical region, we decide to reject the null hypothesis that the population proportion is equal to .20 or 20% of the homes.
This problem also involves a hypothesis test involving one population proportion. As we have done in previous problems, we will go through each step of the process. We begin the hypothesis test by stating the null and alternative hypothesis. The null hypothesis is that the proportion of Americans who believe that prisons should be places of punishment rather than rehabilitation is 31% or .31. The alternative hypothesis is that the true population proportion is greater than .31. Since we have stated a direction in our alternative hypothesis, we will conduct a one-tail hypothesis test. The null and alternative hypotheses are:

H₀: p = .31

H₁: p > .31

Now we need to select our test statistic and the sampling distribution of that test statistic. Since this is a problem involving a population proportion with a large sample size (n=110), we can use the z test and the z distribution.

The third step in a hypothesis test is to select a level of significance and determine both the critical value and the critical region of the test statistic. The problem tells you to use an alpha level of .05. Since we are conducting a one-tail hypothesis test that states that the population proportion is greater than some given value, our critical value and critical region lie in the right tail of the z distribution. Going to the z table, with a=.05, we can see that our critical value of z is equal to 1.65. The critical region is the area to the right of this z value (shown in the picture).

Any obtained value of z greater than or equal to 1.65 falls in the critical region, and would lead us to reject the null hypothesis. Our decision rule, therefore, is to reject the null hypothesis if our obtained value of z is 1.65 or greater: reject H₀ if z_obt ³ 1.65.

We are now ready to calculate our obtained z value. We have a population proportion of .31, a sample proportion of .46, and a sample size of 110. We put these numbers into our z test formula and find z_obt:

Our obtained value of z is 3.41. Since this is greater than the critical value of 1.65, and falls in the critical region, we will reject the null hypothesis that the population proportion is equal to .31 or 31%.

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