Exam 15: Inference About Population Variances

Which of the following statements is correct regarding the percentile points of the chi-squared distribution?

The value in an F-distribution with $v _ { 1 } = 3$ and $v _ { 2 } = 7$ degrees of freedom such that the area to its left is 0.99 is 0.036.

Which of the following is the most common null hypothesis used when testing for equality of two population variances?

Which of the following best describes the sampling distribution of s₁²/s₂² , if we have independently sampled from two normal populations?

A study wants to investigate whether the population variance is greater than 8, if a random sample of size 40, yielded a variance of 10. Which of the following are the correct null hypothesis and alternative hypotheses?

Which of the following best describes the Chi-square distribution?

The value in a chi-squared distribution with 8 degrees of freedom such that the area to its left is 0.95 is 15.5073.

Two independent samples are drawn from two normal populations, where the population variances are assumed to be equal. The sampling distribution of the ratio of the two sample variances is:

The value of $x ^ { 2 }$ with $v$ degrees of freedom, such that the area to its left under the chi-squared curve is equal to A, is denoted by $\bar { x } _ { A , v } ^ { 2 }$ , while $x _ { 1 - A , v } ^ { 2 }$ denotes the value such that the area to its right is A.

Random samples from two normal populations produced the following statistics: $n _ { 1 } =$ 25, $s _ { 1 } ^ { 2 } =$ 75. $n _ { 2 } =$ 13, $S _ { 2 } ^ { 2 } =$ 130. Briefly describe the 95% confidence the ratio of the two population variances: LCL = ( $s _ { 1 } ^ { 2 } /$ $S _ { 2 } ^ { 2 }$ ) / F_0.025,24,12 = 0.191. UCL = ( $s _ { 1 } ^ { 2 } /$ $S _ { 2 } ^ { 2 }$ )F_0.025,12,24 = 1.465.

The ratio of two independent chi-squared variables, each divided by its number of degrees of freedom, is:

The F-distribution is the sampling distribution of the ratio of:

In constructing a 95% interval estimate for the ratio of two population variances, $\sigma _ { 1 } ^ { 2 }$ / $\sigma _ { 2 } ^ { 2 }$ , two independent samples of sizes 41 and 61 are drawn from the populations. If the sample variances are 515 and 920, then the upper confidence limit is:

The value in an F-distribution with $v _ { 1 } = 5$ and $v _ { 2 } = 10$ degrees of freedom such that the area to its left is 0.95 is 4.74.

When the necessary conditions are met, a two-tail test is being conducted at $\alpha$ = 0.05 to test $H _ { 0 } : \sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } = 1$ . The two sample variances are $s _ { 1 } ^ { 2 } = 500 \text { and } s _ { 2 } ^ { 2 } = 900$ , and the sample sizes are $n _ { 1 } = 21 \text { and } n _ { 2 } = 31$ . The rejection region is F > 2.20 or F < 0.4255.

The test statistic employed to test $H _ { 0 } : \sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } = 1$ is $F = s _ { 1 } ^ { 2 } / s _ { 2 } ^ { 2 }$ , which is F-distributed with $v _ { 1 } = n _ { 1 } - 1 \text { and } v _ { 2 } = n _ { 2 } - 1$ degrees of freedom, provided that the two populations are F-distributed.

What are the rejection regions for each of the following sets of hypotheses? a. Ho: σ₁²/σ₂² = 1 H_A: σ₁²/σ₂² ≠ 1 n₁ = 9 n₂ = 20 α = 0.05 b. Ho: σ₁²/σ₂² = 1 H_A: σ₁²/σ₂² < 1 n₁ = 40 n₂ = 50 α = 0.10 c. Ho: σ₁²/σ₂² = 1 H_A: σ₁²/σ₂² > 1 n₁ = 10 n₂ = 8 α = 0.01

We define $F_A , v _ { 1 } , v _ { 2 }$ as the value of the F with $v _ { 1 }$ and $v _ { 2 }$ degrees of freedom such that the area to its left under the F curve is A, while $F 1 - A , v _ { 1 } , v _ { 2 }$ is defined as the value such that the area to its left is A.

For each of the following hypothesis tests, state for what values of χ² we would reject Ho. a. Ho: σ₁² = 10 HA: σ₁² ≠ 10 α = 0.10 and n = 50 b. Ho: σ₁² = 4 HA: σ₁² > 4 α = 0.05 and n = 30 c. Ho: σ₁² = 0.50 HA: σ₁² < 0.50 α = 0.01 and n = 45

A statistician wants to test for the equality of means in two independent samples drawn from normal populations. However, he will not perform the equal-variance t-test of the difference between the population means if the condition necessary for its use is not satisfied. The data are as follows: Sample 1: 7 9 6 15 7 10 8 12 Sample 2: 2 25 9 15 10 18 5 22 27 3 Estimate with 95% confidence the ratio of the two population variances.