GERTRUDE COX: SPREADING THE GOSPEL ACCORDING TO ST. GERTRUDE
Step 2 The confidence interval for σ is from
χα/22 ãs to
n−1
χ12−α/2 ãs,
whereχ12−α/2andχα/22are found in Step 1,nis the sample size, andsis computed from the sample data obtained.
Step 3 Interpret the confidence interval.
EXAMPLE 11.7 The One-Standard-Deviation χ2-Interval Procedure
Xenical Capsules Use the sample data in Table 11.2 to determine a 90% confidence interval for the standard deviation,σ, of the weights of all Xenical capsules.
Solution We apply Procedure 11.2.
Step 1 For a confidence level of1−α, use Table VII to findχ12−α/2andχα/22 with df=n−1.
For a 90% confidence interval, the confidence level is 0.90=1−0.10, and so α=0.10. Also, forn=10, df=9. In Table VII, we find that
χ12−α/2=χ12−0.10/2=χ02.95=3.325 and
χα/22=χ02.10/2=χ02.05=16.919.
†The one-standard-deviationχ2-interval procedure is also known as theχ2-interval procedure for one popula- tion standard deviation.This confidence-interval procedure is often formulated in terms of variance instead of standard deviation.
Step 2 The confidence interval forσ is from n−1
χα/22 ãs to
n−1 χ12−α/2 ãs.
We haven=10, and from Step 1,χ12−α/2=3.325 andχα/22=16.919. Also, we found in Example 11.4 thats =1.055 mg. So, a 90% confidence interval forσ is
from
10−1
16.919ã1.055 to
10−1
3.325 ã1.055, or 0.77 to 1.74.
Step 3 Interpret the confidence interval.
Interpretation We can be 90% confident that the standard deviation of the weights of all Xenical capsules is somewhere between 0.77 mg and 1.74 mg.
Report 11.2
Exercise 11.27 on page 547
THE TECHNOLOGY CENTER
Some statistical technologies have programs that automatically perform one-standard- deviationχ2-procedures, but others do not. In this subsection, we present output and step-by-step instructions for such programs.
Note to Excel users:At the time of this writing, Excel does not have a built-in program for one-standard-deviationχ2-procedures.
Note to TI-83/84 Plus users:
r At the time of this writing, the TI-83/84 Plus does not have a built-in program for one-standard-deviation χ2-procedures. However, TI programs called STDEVHT and STDEVINT, supplied in the TI Programs section on the WeissStats site, allow you to perform those procedures. Your instructor can show you how to download the programs to your calculator.
r Warning:Any data that you may have previously stored in List 1 will be erased dur- ing program execution, so copy those data to another list prior to program execution if you want to retain them.
EXAMPLE 11.8 Using Technology to Conduct One-Standard-Deviation χ2-Procedures
Xenical Capsules Table 11.2 on page 542 gives the weights, in milligrams, of a sample of 10 Xenical capsules. Use Minitab or the TI-83/84 Plus to perform the hypothesis test in Example 11.6 and obtain the confidence interval in Example 11.7.
Solution Letσ denote the population standard deviation of weights of all Xenical capsules. The task in Example 11.6 is to perform the hypothesis test
H0: σ =2.0 mg (too much weight variation) Ha: σ <2.0 mg (not too much weight variation)
at the 5% significance level; the task in Example 11.7 is to find a 90% confidence interval forσ. We applied the appropriate Minitab and TI-83/84 Plus programs to the data, resulting in Output 11.2. Steps for generating that output are presented in Instructions 11.1.Note to Minitab users: For brevity, we have presented only the essential portions of the actual output.
11.1 Inferences for One Population Standard Deviation∗ 545
MINITAB
OUTPUT 11.2 One-standard-deviationχ2-test and interval on the weight data
TI-83/84 PLUS
Using theSTDEVHTprogram
Using theSTDEVINTprogram
As shown in Output 11.2, theP-value for the hypothesis test is 0.019. Because theP-value is less than the specified significance level of 0.05, we reject H0. Out- put 11.2 also shows that a 90% confidence interval forσis from 0.77 mg to 1.74 mg.
INSTRUCTIONS 11.1 Steps for generating Output 11.2 MINITAB
Store the data from Table 11.2 in a column named WEIGHT.
FOR THE HYPOTHESIS TEST:
1 ChooseStat➤Basic Statistic➤1 Variance. . . 2 Press the F3 key to reset the dialog box
3 Click in the text box directly below theOne or more samples, each in a columndrop-down list box and specify WEIGHT
4 Check thePerform hypothesis testcheck box 5 Type2in theValuetext box
6 Click theOptions. . . button
7 Click the arrow button at the right of theAlternative hypothesisdrop-down list box and selectStandard deviation<hypothesized standard deviation 8 ClickOKtwice
FOR THE CI:
1 Repeat steps 1–3 from the hypothesis-test instructions 2 Click theOptions. . . button
3 Click in theConfidence leveltext box and type90 4 ClickOKtwice
TI-83/84 PLUS
Store the data from Table 11.2 in a list named WT.
FOR THE HYPOTHESIS TEST:
1 PressPRGM
2 Arrow down to STDEVHT and pressENTERtwice 3 Type1forTYPEand pressENTER
4 Press2ND➤LIST, arrow down to WT, and press ENTERtwice
5 Type2forSIGMA0and pressENTER 6 Type-1forTYPEand pressENTER FOR THE CI:
1 PressPRGM
2 Arrow down to STDEVINT and pressENTERtwice 3 Type1forTYPEand pressENTER
4 Press2ND➤LIST, arrow down to WT, and press ENTERtwice
5 Type.90forC-LEVELand pressENTER
Exercises 11.1
Understanding the Concepts and Skills
11.1 What is meant by saying that a variable has a chi-square distri- bution?
11.2 How are different chi-square distributions identified?
11.3 Two χ2-curves have degrees of freedom 12 and 20, respec- tively. Which curve more closely resembles a normal curve? Explain your answer.
11.4 The t-table has entries for areas of 0.10, 0.05, 0.025, 0.01, and 0.005. In contrast, theχ2-table has entries for those areas and for 0.995, 0.99, 0.975, 0.95, and 0.90. Explain why thet-values cor- responding to these additional areas can be obtained from the existing t-table but must be provided explicitly in theχ2-table.
In Exercises11.5–11.12, use Table VII to find the requiredχ2-values.
Illustrate your work graphically.
11.5 For aχ2-curve with 19 degrees of freedom, find theχ2-value that has area
a. 0.025 to its right. b. 0.95 to its right.
11.6 For aχ2-curve with 22 degrees of freedom, find theχ2-value that has area
a. 0.01 to its right. b. 0.995 to its right.
11.7 For aχ2-curve with df=10, determine a. χ02.05. b. χ02.975. 11.8 For aχ2-curve with df=4, determine
a. χ0.0052 . b. χ0.992 .
11.9 Consider aχ2-curve with df=8. Obtain theχ2-value that has area a. 0.01 to its left. b. 0.95 to its left.
11.10 Consider aχ2-curve with df=16. Obtain theχ2-value that has area
a. 0.025 to its left. b. 0.975 to its left.
11.11 Determine the twoχ2-values that divide the area under the curve into a middle 0.95 area and two outside 0.025 areas for a χ2-curve with
a. df=5. b. df=26.
11.12 Determine the two χ2-values that divide the area under the curve into a middle 0.90 area and two outside 0.05 areas for a χ2-curve with
a. df=11. b. df=28.
11.13 When you use chi-square procedures to make inferences about a population standard deviation, why should the variable under con- sideration be normally distributed or nearly so?
11.14 Give two situations in which making an inference about a pop- ulation standard deviation would be important.
In each of Exercises11.15–11.20, we have provided a sample stan- dard deviation and sample size. In each case, use the one-standard- deviationχ2-test and the one-standard-deviationχ2-interval proce- dure to conduct the required hypothesis test and obtain the specified confidence interval.
11.15 s=3 andn=10
a. H0:σ =4,Ha:σ <4,α=0.05 b. 90% confidence interval
11.16 s=2 andn=10
a. H0:σ=4,Ha:σ <4,α=0.05 b. 90% confidence interval 11.17 s=7 andn=26
a. H0:σ=5,Ha:σ >5,α=0.01 b. 98% confidence interval 11.18 s=6 andn=26
a. H0:σ=5,Ha:σ >5,α=0.01 b. 98% confidence interval 11.19 s=5 andn=20
a. H0:σ=6,Ha:σ=6,α=0.05 b. 95% confidence interval 11.20 s=8 andn=20
a. H0:σ=6,Ha:σ=6,α=0.05 b. 95% confidence interval
Applying the Concepts and Skills
Preliminary data analyses and other information suggest that you can reasonably assume that the variables under consideration in Exer- cises11.21–11.26are normally distributed. In each case, use either the critical-value approach or the P-value approach to perform the required hypothesis test.
11.21 Grey-Seal Nursing. The average lactation (nursing) period of all earless seals is 23 days. Grey seals are one of several types of earless seals. The length of time that a female grey seal nurses her pup is studied by S. Twiss et al. in the article “Variation in Female Grey Seal (Halichoerus grypus) Reproductive Performance Corre- lates to Proactive-Reactive Behavioural Types” (PLOS ONE 7(11):
e49598. doi:10.1371/journal.pone.0049598). A sample of 14 female grey seals had the following lactation periods, in days.
20.2 20.9 20.6 23.6 19.6 15.9 19.8 15.4 21.4 19.5 17.4 21.9 22.3 16.4
In Exercise 9.84, you were asked to use these data to decide whether the mean lactation period of grey seals differs from 23 days. There, you were to assume that the population standard deviation of lactation periods for grey seals is 3.0 days. At the 10% significance level, do the data provide evidence against that assumption? (Note: s=2.501.) 11.22 EPA Gas Mileage Estimates. Gas mileage estimates for cars and light-duty trucks are determined and published by theU.S. Envi- ronmental Protection Agency(EPA). According to the EPA, “. . . the mileages obtained by most drivers will be within plus or minus 15 percent of the [EPA] estimates. . . .” The mileage estimate given for one model is 23 mpg on the highway. If the EPA claim is true, the stan- dard deviation of mileages should be about 0.15ã23/3=1.15 mpg.
A random sample of 12 cars of this model yields the following high- way mileages.
24.1 23.3 22.5 23.2 22.3 21.1 21.4 23.4 23.5 22.8 24.5 24.3
11.1 Inferences for One Population Standard Deviation∗ 547 a. At the 5% significance level, do the data suggest that the standard
deviation of highway mileages for all cars of this model is differ- ent from 1.15 mpg? (Note: s=1.071.)
b. Why it is useful to know the standard deviation of the gas mileages as well as the mean gas mileage?
11.23 Process Capability. R. Morris and E. Watson studied various aspects of process capability in the paper “Determining Process Capa- bility in a Chemical Batch Process” (Quality Engineering, Vol. 10(2), pp. 389–396). In one part of the study, the researchers compared the variability in product (as measured by standard deviation) of a par- ticular piece of equipment to a known analytic capability to decide whether product consistency could be improved. The following data were obtained for 10 batches of product.
30.1 30.7 30.2 29.3 31.0 29.6 30.4 31.2 28.8 29.8
At the 1% significance level, do the data provide sufficient evidence to conclude that the product variability for this piece of equipment exceeds the analytic capability of 0.27? (Note: s=0.756.)
11.24 Frozen Meals. COOKof Sittingbourne, England, provides baking instructions for its frozen meals. According to the instruc- tions, the average baking time for Chicken and Tomato pasta is 10 to 12 minutes. If the times are normally distributed, the standard de- viation of the times should be approximately 1 minute. A random sample of 15 pastas yielded the following baking times to the nearest tenth of a minute.
10.2 15.1 11.6 13.0 12.4 12.8 10.4 11.5 11.5 11.1 11.6 10.5 12.5 10.1 11.9
At the 1% significance level, do the data provide sufficient evidence to conclude that the standard deviation of baking times exceeds 1 minute? (Note:The sample standard deviation of the 15 baking times is 1.44 minutes.)
11.25 Dispensing Coffee. A coffee machine is supposed to dis- pense 8 fluid ounces (floz) of coffee into a paper cup. In reality, the amounts dispensed vary from cup to cup. However, if the machine is working properly, most of the cups will contain within 15% of the ad- vertised 8 fl oz. In other words, the standard deviation of the amounts dispensed should be less than 0.4 fl oz. A random sample of 15 cups provided the following data, in fluid ounces.
7.24 7.79 8.20 8.09 7.92 8.02 7.64 7.89 8.18 7.80 8.00 7.80 8.14 8.04 8.18
a. At the 5% significance level, do the data provide sufficient evi- dence to conclude that the standard deviation of the amounts being dispensed is less than 0.4 fl oz? (Note: s=0.255.)
b. Why is it important that the standard deviation of the amounts of coffee being dispensed not be too large?
11.26 Counting Production. In Issue 10 ofSTATSfromIowa State University, data were published from an experiment that examined the effects of machine adjustment on bolt production. An electronic counter records the number of bolts passing it on a conveyer belt and stops the run when the count reaches a preset number. The follow- ing data give the times, in seconds, needed to count 20 bolts for eight different runs.
10.78 9.39 9.84 13.94 12.33 7.32 7.91 15.58
Do the data provide sufficient evidence to conclude that the standard deviation in the time needed to count 20 bolts exceeds 2 seconds? Use α=0.05. (Note:The sample standard deviation of the eight times is 2.8875 seconds.)
In Exercises11.27–11.32, use Procedure 11.2 on page 543 to obtain the required confidence interval.
11.27 Grey-Seal Nursing. Refer to Exercise 11.21 and find a 90% confidence interval for the standard deviation of lactation pe- riods of grey seals.
11.28 EPA Gas Mileage Estimates. Refer to Exercise 11.22 and find a 95% confidence interval for the standard deviation of highway gas mileages for all cars of the model in question.
11.29 Process Capability. Refer to Exercise 11.23 and determine a 98% confidence interval for the product variability of the piece of equipment under consideration.
11.30 Frozen Meals. Refer to Exercise 11.24 and determine a 98% confidence interval for the standard deviation of baking times.
11.31 Dispensing Coffee. Refer to Exercise 11.25 and obtain a 99% confidence interval for the standard deviation of the amounts of coffee being dispensed.
11.32 Counting Production. Refer to Exercise 11.26 and obtain a 90% confidence interval for the standard deviation of the times needed to count 20 bolts.
In each of Exercises11.33–11.36, use the technology of your choice to decide whether applying one-standard-deviationχ2-procedures ap- pears reasonable. Explain your answers.
11.33 Military Assistance Loans. The annual update ofU.S. Over- seas Loans and Grants, informally known as the “Greenbook,” con- tains data on U.S. government monetary economic and military as- sistance loans. The following table shows military assistance loans, in thousands of dollars, to a sample of 10 countries, as reported by theU.S. Agency for International Development.
102 280 33 1643 177
69 180 89 205 695
11.34 Positively Selected Genes. R. Nielsen et al. compared 13,731 annotated genes from humans with their chimpanzee orthologs to identify genes that show evidence of positive selection. The re- searchers published their findings in “A Scan for Positively Selected Genes in the Genomes of Humans and Chimpanzees” (PLOS Biology, Vol. 3, Issue 6, pp. 976–985). A simple random sample of 14 tissue types yielded the following number of genes.
14 83 201 36 43 60 70
93 133 33 101 179 134 82
11.35 Total Solar Irradiance. The amount of solar energy trans- mitted through each square meter of space prior to entry in the earth’s atmosphere, as measured by satellites, is called the total solar irradi- ance (TSI). A random sample of daily TSI measurements, in watts per square meter, from theWorld Radiation Centerin Switzerland is as follows.
1365.9587 1365.9485 1365.9308 1365.8971 1365.8925 1365.8803 1365.8829 1365.8680 1365.8817 1365.8631 1365.8836 1365.8846 1365.9098 1365.9680 1365.9091 1365.9294
11.36 Plesiadapis cookei. One extinct relative of primates that lived in North America about 60 million years ago is called Ple- siadapis cookei. Dental characteristics ofP. cookeiwere compared to those of other primate species in the article “Evidence of Di- etary Differentiation among Late Paleocene-Early Eocene Plesi- adapids” (American Journal of Physical Anthropology, Vol. 142, No. 2, pp. 194–210) by D. Boyer et al. The following table gives the dentary depth, in millimeters, for a sample of molars from the skulls of 18P. cookeispecimens.
18.12 19.48 19.36 15.94 15.83 19.70 15.76 17.00 13.96 16.55 15.70 17.83 13.25 16.12 18.13 14.02 14.04 16.20
Working with Large Data Sets
11.37 Body Temperature. A study by researchers at theUniversity of Marylandaddressed the question of whether the mean body tem- perature of humans is 98.6◦F. The results of the study by P. Mack- owiak et al. appeared in the article “A Critical Appraisal of 98.6◦F, the Upper Limit of the Normal Body Temperature, and Other Lega- cies of Carl Reinhold August Wunderlich” (Journal of the American Medical Association, Vol. 268, pp. 1578–1580). Among other data, the researchers obtained the body temperatures of 93 healthy humans, as provided on the WeissStats site. Use the technology of your choice to do the following.
a. Obtain a normal probability plot, boxplot, histogram, and stem- and-leaf diagram of the data.
b. Based on your results from part (a), can you reasonably apply one-standard-deviationχ2-procedures to the data? Explain your reasoning.
c. In Exercise 9.91, you were asked to use these data to de- cide whether mean body temperature of healthy humans differs from 98.6◦F. There, you were to assume that the population stan- dard deviation of body temperatures for healthy humans is 0.63◦F.
At the 5% significance level, do the data provide evidence against that assumption?
d. Find and interpret a 95% confidence interval for the population standard deviation of body temperatures for healthy humans.
11.38 Dexamethasone and IQ. In the paper “Outcomes at School Age After Postnatal Dexamethasone Therapy for Lung Disease of Prematurity” (New England Journal of Medicine, Vol. 350, No. 13, pp. 1304–1313), T. Yeh et al. studied the outcomes at school age in children who had participated in a double-blind, placebo-controlled trial of early postnatal dexamethasone therapy for the prevention of chronic lung disease of prematurity. All of the infants in the study had had severe respiratory distress syndrome requiring mechanical ventilation shortly after birth. On the WeissStats site, we provide the school-age IQs of the 74 children in the control group, based on the study results. Use the technology of your choice to do the following.
a. Obtain a normal probability plot, boxplot, histogram, and stem- and-leaf diagram of the data.
b. Based on your results from part (a), can you reasonably apply one-standard-deviationχ2-procedures to the data? Explain your reasoning.
c. Overall, IQs of school-age children have a standard deviation of 16. At the 1% significance level, do the data provide sufficient evidence to conclude that IQs of school-age children in similar postnatal circumstances as those in the control group of this study have a smaller standard deviation than that of school-age children in general?
d. Find and interpret a 99% confidence interval for the standard deviation of IQs of all school-age children in similar postnatal circumstances as those in the control group of this study.
11.39 Forearm Length. In 1903, K. Pearson and A. Lee published the paper “On the Laws of Inheritance in Man. I. Inheritance of Phys- ical Characters” (Biometrika, Vol. 2, pp. 357–462). The article exam- ined and presented data on forearm length, in inches, for a sample of 140 men, which we provide on the WeissStats site. Use the technol- ogy of your choice to do the following.
a. Obtain a normal probability plot, boxplot, and histogram of the data.
b. Based on your results from part (a), can you reasonably apply one-standard-deviationχ2-procedures to the data? Explain your reasoning.
c. If you answered “yes” to part (b), determine and interpret a 95% confidence interval for the standard deviation of men’s fore- arm length.
Extending the Concepts and Skills
11.40 Xenical Capsules. In Example 11.4 on page 538, we stated that, based on standards set by theUnited States Pharmacopeia(USP), a standard deviation of Xenical capsule weights of less than 2 mg is acceptable. We now ask you to obtain that result. In doing so, we pre- sume that weights of Xenical capsules are normally distributed with a mean of 120 mg.
a. According to USP, the requirements for weight variation of cap- sules are met if each of the individual weights is within the limits of 90% and 110% of the mean weight. Find the lower and upper weight limits in order for USP requirements to be met.
b. Using statistical software, find the percentage of all possible ob- servations of a normally distributed variable that lie within six standard deviations to either side of the mean.
c. Show that, ifσ <2, then fewer than two of every billion Xenical capsules will have weights that violate USP requirements. (Hint:
First determine the value ofσ for which six standard deviations to either side of the mean give the lower and upper weight limits for USP requirements to be met.)
d. Explain why a standard deviation of Xenical capsule weights of less than 2 mg is reasonably acceptable with respect to USP re- quirements.
11.41 Intelligence Quotients. Measured on the Stanford Revision of the Binet–Simon Intelligence Scale, intelligence quotients (IQs) are known to be normally distributed with a mean of 100 and a stan- dard deviation of 16. Use the technology of your choice to do the following.
a. Simulate 1000 samples of four IQs each.
b. Determine the sample standard deviation of each of the 1000 samples.
c. Obtain the following quantity for each of the 1000 samples:
n−1
σ2 s2= 4−1 162 s2.
d. Obtain a histogram of the 1000 values found in part (c).
e. Theoretically, what is the distribution of the variable in part (c)?
f. Compare your answers from parts (d) and (e).