1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach

2 2 Slide IS 310 – Business Statistics Inferences About Population Variances Previously, we have made inferences about population means and proportions. We can make similar inferences about population variances. Examples: A manufacturing plant produces a part whose length should be 1.2”. Some variation of length is allowed. We want to determine that the variation of length is no more than 0.05”. The amount of rainfall varies from state to state. A researcher wants to know if the variation in the amount of rainfall in a specific state is less than 3”. In these cases, we are dealing with population variances (not means or proportions)

3 3 Slide IS 310 – Business Statistics Inferences About Population Variances Inferences About Population Variances n Inference about a Population Variance

4 4 Slide IS 310 – Business Statistics Inferences About a Population Variance n Chi-Square Distribution n Interval Estimation n Hypothesis Testing

5 5 Slide IS 310 – Business Statistics Inferences About a Population Variance In studying population variances, the following quantity is important (n – 1) s / σ (n – 1) s / σ This quantity follows a distribution that is called chi-square distribution (with n – 1 degree of freedom) As the degree of freedom increases with sample size, the shape of the chi- square distribution resembles that of a normal distribution. See Figure 11.1 (10-page 437; 11-page 450)

6 6 Slide IS 310 – Business Statistics Examples of Sampling Distribution of ( n - 1) s 2 /  With 2 degrees of freedom of freedom With 2 degrees of freedom of freedom With 5 degrees of freedom of freedom With 5 degrees of freedom of freedom With 10 degrees of freedom of freedom With 10 degrees of freedom of freedom

7 7 Slide IS 310 – Business Statistics Chi-Square Distribution The chi-square distribution can be used o To estimate a population variance o To estimate a population variance o Perform hypothesis tests about population o Perform hypothesis tests about population variances variances

8 8 Slide IS 310 – Business Statistics Chi-Square Distribution For example, there is a.95 probability of obtaining a  2 (chi-square) value such that For example, there is a.95 probability of obtaining a  2 (chi-square) value such that We will use the notation to denote the value for the chi-square distribution that provides an area of  to the right of the stated value. We will use the notation to denote the value for the chi-square distribution that provides an area of  to the right of the stated value.

9 9 Slide IS 310 – Business Statistics 95% of the possible  2 values 95% of the possible  2 values 22 2 Interval Estimation of  2

10 Slide IS 310 – Business Statistics Interval Estimation of  2 Substituting ( n – 1) s 2 /  2 for the  2 we get Substituting ( n – 1) s 2 /  2 for the  2 we get n Performing algebraic manipulation we get There is a (1 –  ) probability of obtaining a  2 value There is a (1 –  ) probability of obtaining a  2 value such that such that

11 Slide IS 310 – Business Statistics n Interval Estimate of a Population Variance Interval Estimation of  2 where the    values are based on a chi-square distribution with n - 1 degrees of freedom and where 1 -  is the confidence coefficient.

12 Slide IS 310 – Business Statistics Interval Estimation of  n Interval Estimate of a Population Standard Deviation Taking the square root of the upper and lower Taking the square root of the upper and lower limits of the variance interval provides the confidence interval for the population standard deviation.

13 Slide IS 310 – Business Statistics Buyer’s Digest rates thermostats manufactured for home temperature control. In a recent test, 10 thermostats manufactured by ThermoRite were selected and placed in a test room that was maintained at a temperature of 68 o F. The temperature readings of the ten thermostats are The temperature readings of the ten thermostats are shown on the next slide. Interval Estimation of  2 n Example: Buyer’s Digest (A)

14 Slide IS 310 – Business Statistics Interval Estimation of  2 We will use the 10 readings below to We will use the 10 readings below to develop a 95% confidence interval estimate of the population variance. n Example: Buyer’s Digest (A) Temperature Thermostat

15 Slide IS 310 – Business Statistics Interval Estimation of  2 Selected Values from the Chi-Square Distribution Table Our value For n - 1 = = 9 d.f. and  =.05 For n - 1 = = 9 d.f. and  =.05

16 Slide IS 310 – Business Statistics Interval Estimation of  2 22 2 Area in Upper Tail = For n - 1 = = 9 d.f. and  =.05 For n - 1 = = 9 d.f. and  =.05

17 Slide IS 310 – Business Statistics Interval Estimation of  2 Selected Values from the Chi-Square Distribution Table For n - 1 = = 9 d.f. and  =.05 For n - 1 = = 9 d.f. and  =.05 Our value

18 Slide IS 310 – Business Statistics 22 2 Interval Estimation of  2 n - 1 = = 9 degrees of freedom and  =.05 n - 1 = = 9 degrees of freedom and  = Area in Upper Tail =.025 Area in Upper Tail =.025

19 Slide IS 310 – Business Statistics Sample variance s 2 provides a point estimate of  2. Sample variance s 2 provides a point estimate of  2. Interval Estimation of  2.33 <  2 < 2.33 n A 95% confidence interval for the population variance is given by:

20 Slide IS 310 – Business Statistics n Left-Tailed Test Hypothesis Testing About a Population Variance where is the hypothesized value for the population variance Test Statistic Test Statistic Hypotheses Hypotheses

21 Slide IS 310 – Business Statistics n Left-Tailed Test (continued) Hypothesis Testing About a Population Variance Reject H 0 if p -value <  p -Value approach: Critical value approach: Rejection Rule Rejection Rule Reject H 0 if where is based on a chi-square distribution with n - 1 d.f.

22 Slide IS 310 – Business Statistics n Right-Tailed Test Hypothesis Testing About a Population Variance where is the hypothesized value for the population variance Test Statistic Test Statistic Hypotheses Hypotheses

23 Slide IS 310 – Business Statistics n Right-Tailed Test (continued) Hypothesis Testing About a Population Variance Reject H 0 if Reject H 0 if p -value <  where is based on a chi-square distribution with n - 1 d.f. p -Value approach: Critical value approach: Rejection Rule Rejection Rule

24 Slide IS 310 – Business Statistics n Two-Tailed Test Hypothesis Testing About a Population Variance where is the hypothesized value for the population variance Test Statistic Test Statistic Hypotheses Hypotheses

25 Slide IS 310 – Business Statistics n Two-Tailed Test (continued) Hypothesis Testing About a Population Variance Reject H 0 if p -value <  p -Value approach: Critical value approach: Rejection Rule Rejection Rule Reject H 0 if where are based on a chi-square distribution with n - 1 d.f.

26 Slide IS 310 – Business Statistics Recall that Buyer’s Digest is rating Recall that Buyer’s Digest is rating ThermoRite thermostats. Buyer’s Digest gives an “acceptable” rating to a thermo- stat with a temperature variance of 0.5 or less. Hypothesis Testing About a Population Variance n Example: Buyer’s Digest (B) We will conduct a hypothesis test (with We will conduct a hypothesis test (with  =.10) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”.

27 Slide IS 310 – Business Statistics Hypothesis Testing About a Population Variance Using the 10 readings, we will Using the 10 readings, we will conduct a hypothesis test (with  =.10) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”. n Example: Buyer’s Digest (B) Temperature Thermostat

28 Slide IS 310 – Business Statistics n Hypotheses Hypothesis Testing About a Population Variance Reject H 0 if  2 > n Rejection Rule

29 Slide IS 310 – Business Statistics Selected Values from the Chi-Square Distribution Table For n - 1 = = 9 d.f. and  =.10 For n - 1 = = 9 d.f. and  =.10 Hypothesis Testing About a Population Variance Our value

30 Slide IS 310 – Business Statistics 22 2 Area in Upper Tail =.10 Area in Upper Tail =.10 Hypothesis Testing About a Population Variance n Rejection Region Reject H 0

31 Slide IS 310 – Business Statistics n Test Statistic Hypothesis Testing About a Population Variance Because  2 = 12.6 is less than , we cannot Because  2 = 12.6 is less than , we cannot reject H 0. The sample variance s 2 =.7 is insufficient evidence to conclude that the temperature variance for ThermoRite thermostats is unacceptable. n Conclusion The sample variance s 2 = 0.7

32 Slide IS 310 – Business Statistics n Using the p -Value The sample variance of s 2 =.7 is The sample variance of s 2 =.7 is insufficient evidence to conclude that the insufficient evidence to conclude that the temperature variance is unacceptable (>.5). temperature variance is unacceptable (>.5). Because the p –value >  =.10, we Because the p –value >  =.10, we cannot reject the null hypothesis. cannot reject the null hypothesis. The rejection region for the ThermoRite The rejection region for the ThermoRite thermostat example is in the upper tail; thus, the thermostat example is in the upper tail; thus, the appropriate p -value is less than.90 (  2 = 4.168) appropriate p -value is less than.90 (  2 = 4.168) and greater than.10 (  2 = ). and greater than.10 (  2 = ). Hypothesis Testing About a Population Variance A precise p -value can be found using Minitab or Excel. A precise p -value can be found using Minitab or Excel.

33 Slide IS 310 – Business Statistics Sample Problem Problem # 4 (10-Page 443; 11-Page 457) Given: 2 n = 18 s = 0.36 The 90% confidence interval estimate for population variance is: (n – 1) s / Χ ≤ σ ≤ (n – 1) s /Χ (n – 1) s / Χ ≤ σ ≤ (n – 1) s /Χ [(18 – 1) (0.36)/ < < [(18 – 1) (0.36)/8.672) [(18 – 1) (0.36)/ < < [(18 – 1) (0.36)/8.672)

34 Slide IS 310 – Business Statistics Sample Problem Problem # 9 (10-Page 445; 11-Page 459) 2 Given: n = 30 s =  = H : σ ≤ H : σ > a 0 a Test statistic, Χ = (n – 1) s / σ = (29) (.0005) /.0004 = Reject Null Hypothesis if Χ > Χ , , 29 Since Χ (=36.25) < Χ (=42.557), we do not reject the Null Hypothesis 0. 05, , 29 Test indicates that the population variance specification is not being violated.

35 Slide IS 310 – Business Statistics End of Chapter 11