The Chi-Square Distribution (Optional)
Sometimes we can make statistical inferences by using the chi-square distribution. The probability curve of the χ2 (pronounced chi-square) distribution is skewed to the right. Moreover, the exact shape of this probability curve depends on a parameter that is called the number of degrees of freedom (denoted df). Figure 9.14 illustrates chi-square distributions having 2, 5, and 10 degrees of freedom.
Figure 9.14: Chi-Square Distributions with 2, 5, and 10 Degrees of Freedom
Chapter 5
In order to use the chi-square distribution, we employ a chi-square point, which is denoted . As illustrated in the upper portion of Figure 9.15,
is the point on the horizontal axis under the curve of the chi-square distribution that gives a right-hand tail area equal to α. The value of
in a particular situation depends on the right-hand tail area α and the number of degrees of freedom (df) of the chi-square distribution. Values of
are tabulated in a chi-square table. Such a table is given in Table A.17 of Appendix A (pages 877–878); a portion of this table is reproduced as Table 9.4. Looking at the chi-square table, the rows correspond to the appropriate number of degrees of freedom (values of which are listed down the right side of the table), while the columns designate the right-hand tail area α. For example, suppose we wish to find the chi-square point that gives a right-hand tail area of .05 under a chi-square curve having 5 degrees of freedom. To do this, we look in Table 9.4 at the row labeled 5 and the column labeled
. We find that this
point is 11.0705 (see the lower portion of Figure 9.15).
Figure 9.15: Chi-Square Points
Table 9.4: A Portion of the Chi-Square Table
9.7: Statistical Inference for a Population Variance (Optional)
Chapter 9
A vital part of a V6 automobile engine is the engine camshaft. As the camshaft turns, parts of the camshaft make repeated contact with engine lifters and thus must have the appropriate hardness to wear properly. To harden the camshaft, a heat treatment process is used, and a hardened layer is produced on the surface of the camshaft. The depth of the layer is called the hardness depth of the camshaft. Suppose that an automaker knows that the mean and the variance of the camshaft hardness depths produced by its current heat treatment process are, respectively, 4.5 mm and .2209 mm. To reduce the variance of the camshaft hardness depths, a new heat treatment process is designed, and a random sample of n = 30 camshaft hardness depths produced by using the new process has a mean of and a variance of s2 = .0885. In order to attempt to show that the variance, σ2, of the population of all camshaft hardness depths that would be produced by using the new process is less than .2209, we can use the following result:
Statistical Inference for a Populationa Variance
Suppose that s2 is the variance of a sample of n measurements randomly selected from a normally distributed population having variance σ2. The sampling distribution of the statistic (n− 1) s2 /σ2 is a chi-square distribution having n − 1 degrees of freedom. This implies that
1 A 100(1 − α) percent confidence interval for σ2 is
Here and
are the points under the curve of the chi-square distribution having n − 1 degrees of freedom that give right-hand tail areas of, respectively, α/2 and 1 − (α/2).
2 We can test by using the test statistic
Specifically, if we set the probability of a Type I error equal to α, then we can reject H0 in favor of
a
b
c
Here , and
are based on n − 1 degrees of freedom.
The assumption that the sampled population is normally distributed must hold fairly closely for the statistical inferences just given about σ2 to be valid. When we check this assumption in the camshaft situation, we find that a histogram (not given here) of the sample of n = 30 hardness depths is bell-shaped and symmetrical. In order to compute a 95 percent confidence interval for σ2, we note that Table A.17 (pages 877 and 878) tells us that these points—based on n − 1 = 29 degrees of freedom—are
and
(see Figure 9.16). It follows that a 95 percent confidence interval for σ2 is
Figure 9.16: The Chi-Square Points and
This interval provides strong evidence that σ2 is less than .2209.
If we wish to use a hypothesis test, we test the null hypothesis H0: σ2 = .2209 versus the alternative hypothesis Ha: σ2 < .2209. If H0 can be rejected in favor of Ha at the .05 level of significance, we will conclude that the new process has reduced the variance of the camshaft hardness depths. Since the histogram of the sample of n = 30 hardness depths is bell shaped and symmetrical, the appropriate test statistic is given in the summary box. Furthermore, since Ha: σ2 < .2209 is of the form , we should reject H0: σ 2=.2209 if the value of χ2 is less than the critical value
Here
is based on n − 1 = 30 − 1 = 29 degrees of freedom, and this critical value is illustrated in Figure 9.17. Since the sample variance is s2 = .0885, the value of the test statistic is
Figure 9.17: Testing H0: σ2 = .2209 versus Ha: σ2 < .2209 by Setting α = .05
Since χ2 = 11.6184 is less than we reject H0: σ2 = .2209 in favor of Ha: σ2 < .2209. That is, we conclude (at an α of .05) that the new process has reduced the variance of the camshaft hardness depths.
Exercises for Sections 9.6 and 9.7
CONCEPTS
9.84 What assumption must hold to use the chi-square distribution to make statistical inferences about a population variance?
9.85 Define the meaning of the chi-square points and
. Hint: Draw a picture.
9.86 Give an example of a situation in which we might wish to compute a confidence interval for σ2.
METHODS AND APPLICATIONS
Exercises 9.87 through 9.90 relate to the following situation: Consider an engine parts supplier and suppose the supplier has determined that the variance of the population of all cylindrical engine part outside diameters produced by the current machine is approximately equal to, but no less than, .0005. To reduce this variance, a new machine is designed, and a random sample of n = 25 outside diameters produced by this new machine has a mean of and a variance of s2 = .00014. Assume the population of all cylindrical engine part outside diameters that would be produced by the new machine is normally distributed, and let σ2 denote the variance of this population.
9.87 Find a 95 percent confidence interval for σ2.
9.88 Test H0: σ2 = .0005 versus Ha: σ2 < .0005 by setting α = .05.
9.89 Find a 99 percent confidence interval for σ2.
9.90 Test H0: σ2 = .0005 versus Ha: σ2 ≠ .0005 by setting α = .01.
Chapter Summary
We began this chapter by learning about the two hypotheses that make up the structure of a hypothesis test. The null hypothesis is the statement being tested. Usually it represents the status quo and it is not rejected unless there is convincing sample evidence that it is false. The alternative, or, research, hypothesis is a statement that is accepted only if there is convincing sample evidence that it is true and that the null hypothesis is false. In some situations, the alternative hypothesis is a condition for which we need to attempt to find supportive evidence. We also learned that two types of errors can be made in a hypothesis test. A Type I error occurs when we reject a true null hypothesis, and a Type II error occurs when we do not reject a false null hypothesis.
We studied two commonly used ways to conduct a hypothesis test. The first involves comparing the value of a test statistic with what is called a critical value, and the second employs what is called a p-value. The p-value measures the weight of evidence against the null hypothesis. The smaller the p-value, the more we doubt the null hypothesis. We learned that, if we can reject the null hypothesis with the probability of a Type I error equal to α, then we say that the test result has statistical significance at the α level. However, we also learned that, even if the result of a hypothesis test tells us that statistical significance exists, we must carefully assess whether the result is practically important. One good way to do this is to use a point estimate and confidence interval for the parameter of interest.
The specific hypothesis tests we covered in this chapter all dealt with a hypothesis about one population parameter. First, we studied a test about a population mean that is based on the assumption that the population standard deviation σ is known. This test employs the normal distribution. Second, we studied a test about a population mean that assumes that σ is unknown. We learned that this test is based on the t distribution. Figure 9.18 presents a flowchart summarizing how to select an appropriate test statistic to test a hypothesis about a population mean. Then we presented a test about a population proportion that is based on the normal distribution. Next (in optional Section 9.5) we studied Type II error probabilities, and we showed how we can find the sample size needed to make both the probability of a Type I error and the probability of a serious Type II error as small as we wish. We concluded this chapter by discussing (in optional Sections 9.6 and 9.7) the chi-square distribution and its use in making statistical inferences about a population variance.
Figure 9.18: Selecting an Appropriate Test Statistic to Test a Hypothesis about a Population Mean
Glossary of Terms
alternative (research) hypothesis:
A statement that will be accepted only if there is convincing sample evidence that it is true. Sometimes it is a condition for which we need to attempt to find supportive evidence. (page 347)
chi-square distribution:
A useful continuous probability distribution. Its probability curve is skewed to the right, and the exact shape of the probability curve depends on the number of degrees of freedom associated with the curve. (page 382)
critical value:
The value of the test statistic is compared with a critical value in order to decide whether the null hypothesis can be rejected. (pages 354, 358, 360)
greater than alternative:
An alternative hypothesis that is stated as a greater than ( > ) inequality. (page 349)
less than alternative:
An alternative hypothesis that is stated as a less than ( < ) inequality. (page 349)
not equal to alternative:
An alternative hypothesis that is stated as a not equal to ( ≠ ) inequality. (page 349)
null hypothesis:
The statement being tested in a hypothesis test. It usually represents the status quo and it is not rejected unless there is convincing sample evidence that it is false. (page 347)
one-sided alternative hypothesis:
An alternative hypothesis that is stated as either a greater than ( > ) or a less than ( < ) inequality. (page 349)
power (of a statistical test):
The probability of rejecting the null hypothesis when it is false. (page 379)
p-value (probability value):
The probability, computed assuming that the null hypothesis is true, of observing a value of the test statistic that is at least as extreme as the value actually computed from the sample data. The p-value measures how much doubt is cast on the null hypothesis by the sample data. The smaller the p-value, the more we doubt the null hypothesis. (pages 355, 358, 360, 362)
statistical significance at the α level:
When we can reject the null hypothesis by setting the probability of a Type I error equal to α. (page 354)
test statistic:
A statistic computed from sample data in a hypothesis test. It is either compared with a critical value or used to compute a p-value. (page 349)
two-sided alternative hypothesis:
An alternative hypothesis that is stated as a not equal to ( ≠ ) inequality. (page 349)
Type I error:
Rejecting a true null hypothesis. (page 350)
Type II error:
Failing to reject a false null hypothesis. (page 350)
Important Formulas and Tests
Hypothesis Testing steps: page 357
A hypothesis test about a population mean (σ known): page 361
A t test about a population mean (σ unknown): page 366
A large sample hypothesis test about a population proportion: page 371
Calculating the probability of a Type II error: page 379
Sample size determination to achieve specified values of α and β: page 380
Statistical inference about a population variance: page 383
Supplementary Exercises