Statistical Inferences for Population Variances fileStatistical Inference for Population Variance...

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Statistical Inferences for Population Variances

Transcript of Statistical Inferences for Population Variances fileStatistical Inference for Population Variance...

Page 1: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

Statistical Inferences for Population Variances

Page 2: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

The Chi-Square Distribution

• Sometimes make inferences using the chi-square distribution• Denoted ²

• Skewed to the right

• Exact shape depends on the degrees of freedom• Denoted df

•A chi-square point ²α is the point under a chi-square distribution that gives right-hand tail area

Page 3: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

The Chi-Square Distribution Continued

Page 4: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

A Portion of the Chi-Square Table

Page 5: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

Statistical Inference for Population Variance

• If s2 is the variance of a random sample of n measurements from a normal population with variance σ2

• The sampling distribution of the statistic(n - 1) s2 / σ2 is a chi-square distribution with (n – 1) degrees of freedom

• Can calculate confidence interval and perform hypothesis testing

• 100(1-α)% confidence interval for σ2

Page 6: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

Formulas

2

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22

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0

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2

2/1

2

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2

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statistic test theusing at :Hcan test We

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Page 7: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

Hypothesis Testing for Population Variance

Page 8: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

F Distribution

Figure 11.5

Page 9: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

F Distribution Tables

• The F point F is the point on the horizontal axis under the curve of

the F distribution that gives a right-hand tail area equal to • The value of F depends on a (the size of the right-hand tail area)

and df1 and df2

• Different F tables for different values of •Tables A.6 for = 0.10•Tables A.7 for = 0.05•Tables A.8 for = 0.025•Tables A.9 for = 0.01

Page 10: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

A Portion of an F Table: Values of F.05

Table 11.2

Page 11: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

Comparing Two Population Variances by Using Independent Samples

• Population 1 has variance σ12 and population 2 has

variance σ22

• The null hypothesis H0 is that the variances are the same• H0: σ1

2 = σ22

• The alternative is that one is smaller than the other• That population has less variable measurements• Suppose σ1

2 > σ22

• More usual to normalize

• Test H0: σ12/σ2

2 = 1 vs. σ12/σ2

2 > 1

Page 12: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

Comparing Two Population Variances Continued

• Reject H0 in favor of Ha if s12/s2

2 is significantly greater than 1

• s12 is the variance of a random of size n1 from a

population with variance σ12

• s22 is the variance of a random of size n2 from a

population with variance σ22

• To decide how large s12/s2

2 must be to reject H0, describe the sampling distribution of s1

2/s22

• The sampling distribution of s12/s2

2 is the F distribution

Page 13: Statistical Inferences for Population Variances fileStatistical Inference for Population Variance •If s2 is the variance of a random sample of n measurements from a normal population

Two Tailed Alternative

• The null and alternative hypotheses are:• H0: σ1

2= σ2

2

H0: σ12≠ σ2

2

• Test statistic F is the ratio of the larger sample variance divided by the smaller sample variance

• df1 = n-1 for sample having the larger variance and df2 = n-2 for smaller variance

• Reject if F > Fα/2 orp-value < α