GERTRUDE COX: SPREADING THE GOSPEL ACCORDING TO ST. GERTRUDE
11.1 Inferences for One Population Standard Deviation ∗
Recall that standard deviation is a measure of the variation (or spread) of a data set.
Also recall that, for a variablex, the standard deviation of all possible observations for the entire population is called thepopulation standard deviationorstandard deviation of the variable x. It is denotedσx or, when no confusion will arise, simplyσ.
Suppose that we want to obtain information about a population standard deviation.
If the population is small, we can often determineσ exactly by first taking a census and then computing σ from the population data. However, if the population is large, which is usually the case, a census is generally not feasible, and we must use inferential methods to obtain the required information aboutσ.
In this section, we describe how to perform hypothesis tests and construct confi- dence intervals for the standard deviation of a normally distributed variable. Such in- ferences are based on a distribution called thechi-square distribution.Chi (pronounced
“k¯i”) is a Greek letter whose lowercase form isχ. The Chi-Square Distribution
A variable has a chi-square distribution if its distribution has the shape of a spe- cial type of right-skewed curve, called a chi-square (χ2) curve.Actually, there are infinitely many chi-square distributions, and we identify the chi-square distribution (and χ2-curve) in question by its number of degrees of freedom, just as we did for t-distributions. Figure 11.1 shows threeχ2-curves and illustrates some basic proper- ties ofχ2-curves.
FIGURE 11.1 χ2-curves for df = 5, 10, and 19
df = 19
2 df = 5
df = 10
0 5 10 15 20 25 30
KEY FACT 11.1 Basic Properties of χ2-Curves
Property 1: The total area under aχ2-curve equals 1.
Property 2: Aχ2-curve starts at 0 on the horizontal axis and extends indef- initely to the right, approaching, but never touching, the horizontal axis as it does so.
Property 3: Aχ2-curve is right skewed.
Property 4: As the number of degrees of freedom becomes larger, χ2- curves look increasingly like normal curves.
11.1 Inferences for One Population Standard Deviation∗ 537 Percentages (and probabilities) for a variable having a chi-square distribution are equal to areas under its associatedχ2-curve. To perform a hypothesis test or construct a confidence interval for a population standard deviation, we need to know how to find theχ2-value that corresponds to a specified area under aχ2-curve. Table VII in Appendix A providesχ2-values corresponding to several areas for various degrees of freedom.
The χ2-table (Table VII) is similar to the t-table (Table IV). The two outside columns of Table VII, labeled df, display the number of degrees of freedom. As ex- pected, the symbolχα2denotes theχ2-value having areaαto its right under aχ2-curve.
Thus the column headedχ02.995, for example, containsχ2-values having area 0.995 to their right.
EXAMPLE 11.1 Finding the χ2-Value Having a Specified Area to Its Right For aχ2-curve with 12 degrees of freedom, findχ02.025; that is, find theχ2-value having area 0.025 to its right, as shown in Fig. 11.2(a).
FIGURE 11.2 Finding theχ2-value having area 0.025 to its right
2
2-curve df = 12
Area = 0.025 0
(a) (b)
2
2-curve df = 12
Area = 0.025 0
= 23.337
0.0252
0.0252 = ?
Solution To find thisχ2-value, we use Table VII. The number of degrees of free- dom is 12, so we first go down the outside columns, labeled df, to “12.” Then, going across that row to the column labeledχ02.025, we reach 23.337. This number is the χ2-value having area 0.025 to its right, as shown in Fig. 11.2(b). In other words, for aχ2-value with df=12,χ02.025=23.337.
Exercise 11.5 on page 546
EXAMPLE 11.2 Finding the χ2-Value Having a Specified Area to Its Left
Determine theχ2-value having area 0.05 to its left for aχ2-curve with df=7, as depicted in Fig. 11.3(a).
FIGURE 11.3 Finding theχ2-value having area 0.05 to its left
2
2- curve df = 7
0
2= ?
(a) 0.05
2= 2.167
2
2- curve df = 7
0
(b) 0.05
Exercise 11.9 on page 546
Solution Because the total area under a χ2-curve equals 1 (Property 1 of Key Fact 11.1), the unshaded area in Fig. 11.3(a) must equal 1−0.05=0.95. Thus the requiredχ2-value is χ02.95. From Table VII with df= 7,χ02.95=2.167. So, for a χ2-curve with df=7, theχ2-value having area 0.05 to its left is 2.167, as shown in Fig. 11.3(b).
EXAMPLE 11.3 Finding the χ2-Values for a Specified Area
For a χ2-curve with df = 20, determine the two χ2-values that divide the area under the curve into a middle 0.95 area and two outside 0.025 areas, as shown in Fig. 11.4(a).
FIGURE 11.4 Finding the twoχ2-values that divide the area under the curve into a middle 0.95 area and two outside 0.025 areas
0.025
2= ? 34.170
0.025
2
2- curve df = 20
0
2= ?
(a) 0.025
9.591
2
2- curve df = 20
0
(b)
0.025 0.95
0.95
Solution First, we find theχ2-value on the right in Fig. 11.4(a). Because the shaded area on the right is 0.025, theχ2-value on the right isχ02.025. From Table VII with df=20,χ02.025=34.170.
Next, we find theχ2-value on the left in Fig. 11.4(a). Because the area to the left of thatχ2-value is 0.025, the area to its right is 1−0.025=0.975. Hence the χ2-value on the left isχ02.975, which, by Table VII, equals 9.591 for df=20.
Consequently, for aχ2-curve with df=20, the twoχ2-values that divide the area under the curve into a middle 0.95 area and two outside 0.025 areas are 9.591 and 34.170, as shown in Fig. 11.4(b).
Exercise 11.11 on page 546
The Logic Behind Hypothesis Tests for One Population Standard Deviation
We illustrate the logic behind hypothesis tests for one population standard deviation in the next example.
EXAMPLE 11.4 Hypothesis Tests for a Population Standard Deviation
Xenical Capsules Xenical is used to treat obesity in people with risk factors such as diabetes, high blood pressure, and high cholesterol or triglycerides. Xenical works in the intestines, where it blocks some of the fat a person eats from being absorbed.
A standard prescription of Xenical is given in 120-milligram (mg) capsules.
Although the capsule weights can vary somewhat from 120 mg and also from each other, keeping the variation small is important for various medical reasons.
Based on standards set by the United States Pharmacopeia (USP)—an of- ficial public standards-setting authority for all prescription and over-the-counter medicines and other health care products manufactured or sold in the United States—we determined that a standard deviation of Xenical capsule weights of less than 2 mg is acceptable.†
a. Formulate statistically the problem of deciding whether the standard deviation of Xenical capsule weights is less than 2.0 mg.
†See Exercise 11.40 for an explanation of how that information could be obtained.
11.1 Inferences for One Population Standard Deviation∗ 539 b. Explain the basic idea for carrying out the hypothesis test.
TABLE 11.1 Weights (mg) of 10 Xenical capsules
120.94 118.58 119.41 120.23 121.13 118.22 119.71 121.09 120.56 119.11
c. In the paper “HPLC Analysis of Orlistat and Its Application to Drug Quality Control Studies” (Chemical & Pharmaceutical Bulletin, Vol. 55, No. 2, pp. 251–
254), E. Souri et al. studied various properties of Xenical. A sample of 10 Xeni- cal capsules had the weights shown in Table 11.1. Discuss the use of these data to make a decision concerning the hypothesis test.
Solution
a. We want to perform the hypothesis test
H0: σ =2.0 mg (too much weight variation) Ha: σ <2.0 mg (not too much weight variation).
If the null hypothesis can be rejected, we can be confident that the variation in capsule weights is acceptable.†
b. Roughly speaking, the hypothesis test can be carried out in the following manner: