Pendugaan Parameter Varians dan Rasio Varians Pertemuan 18 Matakuliah: I0134/Metode Statistika Tahun: 2007

Bina Nusantara Slides Prepared by JOHN S. LOUCKS St. Edward’s University © 2002 South-Western/Thomson Learning 

Bina Nusantara Chapter 11 Inferences About Population Variances Inference about a Population Variance Inferences about the Variances of Two Populations

Bina Nusantara Inferences About a Population Variance Chi-Square Distribution Interval Estimation of  2 Hypothesis Testing

Bina Nusantara Chi-Square Distribution The chi-square distribution is the sum of squared standardized normal random variables such as (z 1 ) 2 +(z 2 ) 2 +(z 3 ) 2 and so on. The chi-square distribution is based on sampling from a normal population. The sampling distribution of (n - 1)s 2 /  2 has a chi-square distribution whenever a simple random sample of size n is selected from a normal population. We can use the chi-square distribution to develop interval estimates and conduct hypothesis tests about a population variance.

Bina Nusantara Interval Estimation of  2 Interval Estimate of a Population Variance where the    values are based on a chi-square distribution with n - 1 degrees of freedom and where 1 -  is the confidence coefficient.

Bina Nusantara Interval Estimation of  Interval Estimate of a Population Standard Deviation Taking the square root of the upper and lower limits of the variance interval provides the confidence interval for the population standard deviation.

Bina Nusantara Chi-Square Distribution With Tail Areas of % of the possible  2 values 95% of the possible  2 values 22 2 Interval Estimation of  2

Bina Nusantara Example: Buyer’s Digest 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 listed below. We will use the 10 readings to develop a 95% confidence interval estimate of the population variance. Therm Temp

Bina Nusantara Example: Buyer’s Digest Interval Estimation of  2 n - 1 = = 9 degrees of freedom and  =.05 22 2

Bina Nusantara Interval Estimation of  2 n - 1 = = 9 degrees of freedom and  =.05 22 2 Example: Buyer’s Digest Area in Upper Tail =.975

Bina Nusantara Example: Buyer’s Digest Interval Estimation of  2 n - 1 = = 9 degrees of freedom and  =.05 22 22 0 0 Area in Upper Tail =.025 Area in Upper Tail =

Bina Nusantara Interval Estimation of  2 Sample variance s 2 provides a point estimate of  2. A 95% confidence interval for the population variance is given by:.33 <  2 < 2.33 Example: Buyer’s Digest

Bina Nusantara Left-Tailed Test – Hypotheses – Test Statistic – Rejection Rule Reject H 0 if (where is based on a chi-square distribution with n - 1 d.f.) or Reject H 0 if p-value <  Hypothesis Testing About a Population Variance

Bina Nusantara n Right-Tailed Test Hypotheses Hypotheses Test Statistic Test Statistic Rejection Rule Rejection Rule Reject H 0 if (where is based on a chi-square distribution with n - 1 d.f.) or Reject H 0 if p -value <  Hypothesis Testing About a Population Variance

Bina Nusantara n Two-Tailed Test Hypotheses Hypotheses Test Statistic Test Statistic Rejection Rule Rejection Rule Reject H 0 if (where are based on a chi-square distribu- tion with n - 1 d.f.) or Reject H 0 if p -value <  Hypothesis Testing About a Population Variance

Bina Nusantara n One-Tailed Test Hypotheses Hypotheses Test Statistic Test Statistic Rejection Rule Rejection Rule Reject H 0 if F > F  where the value of F  is based on an F distribution with n (numerator) and n (denominator) d.f. Hypothesis Testing About the Variances of Two Populations

Bina Nusantara n Two-Tailed Test Hypotheses Hypotheses Test Statistic Test Statistic Rejection Rule Rejection Rule Reject H 0 if F > F  /2 where the value of F  /2 is based on an F distribution with n (numerator) and n (denominator) d.f. Hypothesis Testing About the Variances of Two Populations

Bina Nusantara Buyer’s Digest has conducted the same test, as was described earlier, on another 10 thermostats, this time manufactured by TempKing. The temperature readings of the ten thermostats are listed below. We will conduct a hypothesis test with  =.10 to see if the variances are equal for ThermoRite’s thermostats and TempKing’s thermostats. Therm Temp Example: Buyer’s Digest

Bina Nusantara Hypothesis Testing About the Variances of Two Populations – Hypotheses (ThermoRite and TempKing thermo- stats have same temperature variance) (Their variances are not equal) – Rejection Rule The F distribution table shows that with  =.10, 9 d.f. (numerator), and 9 d.f. (denominator), F.05 = Reject H 0 if F > 3.18 Example: Buyer’s Digest

Bina Nusantara Hypothesis Testing About the Variances of Two Populations – Test Statistic ThermoRite’s sample variance is.70. TempKing’s sample variance is F = 1.52/.70 = 2.17 – Conclusion We cannot reject H 0. There is insufficient evidence to conclude that the population variances differ for the two thermostat brands. Example: Buyer’s Digest

Bina Nusantara End of Chapter 11