Inferences About Population VariancesInference about a Population VarianceInferences about the Variances of Two PopulationsISI:

Inferences About a Population VarianceChi-Square DistributionInterval Estimation of 2Hypothesis Testing

Chi-Square Distribution We can use the chi-square distribution to develop interval estimates and conduct hypothesis tests about a population variance. The sampling distribution of (n - 1)s2/ 2 has a chi- square distribution whenever a simple random sample of size n is selected from a normal population. The chi-square distribution is based on sampling from a normal population. The chi-square distribution is the sum of squared standardized normal random variables such as (z1)2+(z2)2+(z3)2 and so on.

Examples of Sampling Distribution of (n - 1)s2/ 20With 2 degrees of freedomWith 5 degrees of freedomWith 10 degrees of freedom

Chi-Square DistributionFor example, there is a .95 probability of obtaining a c2 (chi-square) value such that

95% of thepossible 2 values20.025.025Interval Estimation of 2

Interval Estimation of 2 Substituting (n 1)s2/s 2 for the c2 we get Performing algebraic manipulation we get There is a (1 a) probability of obtaining a c2 value such that

Interval Estimation of 2Interval Estimate of a Population Variancewhere the values are based on a chi-squaredistribution with n - 1 degrees of freedom andwhere 1 - is the confidence coefficient.

Interval Estimation of Interval Estimate of a Population Standard Deviation Taking the square root of the upper and lowerlimits of the variance interval provides the confidenceinterval for the population standard deviation.

Interval Estimation of 2Buyers Digest rates thermostatsmanufactured for home temperaturecontrol. In a recent test, 10 thermostatsmanufactured by ThermoRite wereselected and placed in a test room thatwas maintained at a temperature of 68oF. The temperature readings of the ten thermostats areshown on the next slide. Example: Buyers Digest (A)

Interval Estimation of 2 We will use the 10 readings below todevelop a 95% confidence intervalestimate of the population variance.Example: Buyers Digest (A)Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2Thermostat 1 2 3 4 5 6 7 8 9 10

Interval Estimation of 2Selected Values from the Chi-Square Distribution Table For n - 1 = 10 - 1 = 9 d.f. and a = .05

Sheet1

DegreesArea in Upper Tail

of Freedom.99.975.95.90.10.05.025.01

50.5540.8311.1451.6109.23611.07012.83215.086

60.8721.2371.6352.20410.64512.59214.44916.812

71.2391.6902.1672.83312.01714.06716.01318.475

81.6472.1802.7333.49013.36215.50717.53520.090

92.0882.7003.3254.16814.68416.91919.02321.666

102.5583.2473.9404.86515.98718.30720.48323.209

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Interval Estimation of 220.025Area inUpper Tail= .9752.700 For n - 1 = 10 - 1 = 9 d.f. and a = .05

Interval Estimation of 2Selected Values from the Chi-Square Distribution Table For n - 1 = 10 - 1 = 9 d.f. and a = .05

Sheet1

DegreesArea in Upper Tail

of Freedom.99.975.95.90.10.05.025.01

50.5540.8311.1451.6109.23611.07012.83215.086

60.8721.2371.6352.20410.64512.59214.44916.812

71.2391.6902.1672.83312.01714.06716.01318.475

81.6472.1802.7333.49013.36215.50717.53520.090

92.0882.7003.3254.16814.68416.91919.02321.666

102.5583.2473.9404.86515.98718.30720.48323.209

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20.0252.700Interval Estimation of 2 n - 1 = 10 - 1 = 9 degrees of freedom and a = .0519.023Area in UpperTail = .025

Interval Estimation of 2Sample variance s2 provides a point estimate of 2..33 < 2 < 2.33A 95% confidence interval for the population variance is given by:

Hypothesis TestingAbout a Population Variance Left-Tailed TestTest StatisticHypotheses

Left-Tailed Test (continued)Hypothesis Testing About a Population VarianceReject H0 if p-value < ap-Value approach:Critical value approach:Rejection Rule

Right-Tailed TestHypothesis TestingAbout a Population VarianceTest StatisticHypotheses

Right-Tailed Test (continued)Hypothesis Testing About a Population VarianceReject H0 if p-value < ap-Value approach:Critical value approach:Rejection Rule

Two-Tailed TestHypothesis TestingAbout a Population VarianceTest StatisticHypotheses

Two-Tailed Test (continued)Hypothesis Testing About a Population VarianceReject H0 if p-value < ap-Value approach:Critical value approach:Rejection Rule

Recall that Buyers Digest is ratingThermoRite thermostats. Buyers Digestgives an acceptable rating to a thermo-stat with a temperature variance of 0.5or less.Hypothesis Testing About a Population VarianceExample: Buyers Digest (B) We will conduct a hypothesis test (witha = .10) to determine whether the ThermoRitethermostats temperature variance is acceptable.

Hypothesis Testing About a Population Variance Using the 10 readings, we willconduct a hypothesis test (with a = .10)to determine whether the ThermoRitethermostats temperature variance isacceptable.Example: Buyers Digest (B)Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2Thermostat 1 2 3 4 5 6 7 8 9 10

HypothesesHypothesis Testing About a Population VarianceReject H0 if c 2 > 14.684 Rejection Rule

Selected Values from the Chi-Square Distribution Table For n - 1 = 10 - 1 = 9 d.f. and a = .10Hypothesis Testing About a Population Variance

Sheet1

DegreesArea in Upper Tail

of Freedom.99.975.95.90.10.05.025.01

50.5540.8311.1451.6109.23611.07012.83215.086

60.8721.2371.6352.20410.64512.59214.44916.812

71.2391.6902.1672.83312.01714.06716.01318.475

81.6472.1802.7333.49013.36215.50717.53520.090

92.0882.7003.3254.16814.68416.91919.02321.666

102.5583.2473.9404.86515.98718.30720.48323.209

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2014.684Area in UpperTail = .10Hypothesis Testing About a Population VarianceRejection RegionReject H0

Test StatisticHypothesis Testing About a Population Variance Because c2 = 12.6 is less than 14.684, we cannotreject H0. The sample variance s2 = .7 is insufficientevidence to conclude that the temperature variancefor ThermoRite thermostats is unacceptable. ConclusionThe sample variance s 2 = 0.7

Using Excel to Conduct a Hypothesis Testabout a Population VarianceUsing the p-Value The sample variance of s 2 = .7 is insufficient evidence to conclude that the temperature variance is unacceptable (>.5). Because the p value > a = .10, we cannot reject the null hypothesis. The rejection region for the ThermoRite thermostat example is in the upper tail; thus, the appropriate p-value is less than .90 (c 2 = 4.168) and greater than .10 (c 2 = 14.684).

One-Tailed TestTest StatisticHypothesesHypothesis Testing About the Variances of Two PopulationsDenote the population providing thelarger sample variance as population 1.

One-Tailed Test (continued)Reject H0 if p-value < awhere the value of F is based on anF distribution with n1 - 1 (numerator)and n2 - 1 (denominator) d.f.p-Value approach:Critical value approach:Rejection RuleHypothesis Testing About the Variances of Two PopulationsReject H0 if F > F

Two-Tailed TestTest StatisticHypothesesHypothesis Testing About the Variances of Two PopulationsDenote the population providing thelarger sample variance as population 1.

Two-Tailed Test (continued)Reject H0 if p-value < ap-Value approach:Critical value approach:Rejection RuleHypothesis Testing About the Variances of Two PopulationsReject H0 if F > F/2where the value of F/2 is based on anF distribution with n1 - 1 (numerator)and n2 - 1 (denominator) d.f.

Buyers Digest has conducted thesame test, as was described earlier, onanother 10 thermostats, this timemanufactured by TempKing. Thetemperature readings of the tenthermostats are listed on the next slide. Hypothesis Testing About the Variances of Two PopulationsExample: Buyers Digest (C) We will conduct a hypothesis test with = .10 to seeif the variances are equal for ThermoRites thermostatsand TempKings thermostats.

Hypothesis Testing About the Variances of Two PopulationsExample: Buyers Digest (C)ThermoRite SampleTempKing SampleTemperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2Thermostat 1 2 3 4 5 6 7 8 9 10 Temperature 67.7 66.4 69.2 70.1 69.5 69.7 68.1 66.6 67.3 67.5Thermostat 1 2 3 4 5 6 7 8 9 10

HypothesesHypothesis Testing About the Variances of Two PopulationsReject H0 if F > 3.18The F distribution table (on next slide) shows that withwith = .10, 9 d.f. (numerator), and 9 d.f. (denominator),F.05 = 3.18.(Their variances are not equal)(TempKing and ThermoRite thermostatshave the same temperature variance)Rejection Rule

Selected Values from the F Distribution TableHypothesis Testing About the Variances of Two Populations

Sheet1

DenominatorArea inNumerator Degrees of Freedom

DegreesUpper

of FreedomTail7891015

8.102.622.592.562.542.46

.053.503.443.393.353.22

.0254.534.434.364.304.10

.016.186.035.915.815.52

9.102.512.472.442.422.34

.053.293.233.183.143.01

.0254.204.104.033.963.77

.015.615.475.355.264.96

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Test StatisticHypothesis Testing About the Variances of Two PopulationsWe cannot reject H0. F = 2.53 < F.05 = 3.18.There is insufficient evidence to conclude thatthe population variances differ for the twothermostat brands.ConclusionTempKings sample variance is 1.768ThermoRites sample variance is .700

Determining and Using the p-ValueHypothesis Testing About the Variances of Two Populations Because a = .10, we have p-value > a and therefore we cannot reject the null hypothesis. But this is a two-tailed test; after doubling the upper-tail area, the p-value is between .20 and .10. Because F = 2.53 is between 2.44 and 3.18, the area in the upper tail of the distribution is between .10 and .05.Area in Upper Tail .10 .05 .025 .01F Value (df1 = 9, df2 = 9) 2.44 3.18 4.03 5.35