Exam 15: Inference About Population Variances

In a random sample of 20 patients who visited the emergency room of hospital 1, a researcher found that the variance of the waiting time (in minutes) was 128.0. In a random sample of 15 patients in the emergency room of hospital 2, the researcher found the variance to be 178.8. Can we infer at the 5% level of significance that the population variances differ?

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 is the correct value of the test statistic? A 50 B 1.25 C 48.75 D 31.2

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 explain how to use the 95% confidence the ratio of the two population variances to test the hypothesis of equal 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.

Which of the following best describes the sampling distribution of s₁²/s₂² , if we have independently sampled from two normal populations? A. Z distribution B. t distribution C. Chi square distribution D. F distribution

Which of the following is the test statistic for σ²? A. Z test statistic B. test statistic C. t test statistic D. None of these choices are correct

A university lecturer wants to investigate if the variance of final marks of students in two of her courses differs. She takes a random sample of 25 students from the mathematics course she lectures and finds the student's final marks had a variance of 5. She takes a random sample of 13 students from the statistics course she lectures and finds the variance of 10. Assuming that the final grades of students in her mathematics and in her statistics course are normally distributed, is there enough evidence at the 5% significance level for this lecturer to infer that the two population variances differ?

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.

Which of the following statements is false? A. The chi-squared distribution is positively skewed. B. The chi-squared distribution is symmetrical. C. All the values of the chi-squared distribution are positive. D. The shape of the chi-squared distribution depends on the number of degrees of freedom.

The value in a chi-squared distribution with 5 degrees of freedom such that the area to its right is 0.10 is 1.61031.

What are the rejection regions for each of the following sets of hypotheses? $\sigma _ { 1 } / \sigma _ { 2 } ^ { 2 } = 1$ $H _ { A } : \sigma _ { 1 } / \sigma _ { 2 } { } ^ { 2 } \neq 1$ $\mathrm { n } _ { 1 } = 9 \quad \mathrm { n } _ { 2 } = 20 \quad \alpha = 0.05$ b. $\quad \mathrm { H_0 } : \sigma _ { 1 } 2 / \sigma _ { 2 } { } ^ { 2 } = 1$ $\mathrm { H } _ { \mathrm {A } } : \sigma _ { 1 } { } ^ { 2 } / \sigma _ { 2 } { } ^ { 2 } < 1$ $\mathrm { n } _ { 1 } = 40 \quad \mathrm { n } _ { 2 } = 50 \quad \alpha = 0.10$ c. $\quad \mathrm { Ho } : \sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } = 1$ $\mathrm { H } _ { \mathrm { A} } : \sigma _ { 1 } 2 / \sigma _ { 2 } ^ { 2 } > 1$ =10 =8 \alpha=0.01

When comparing two population variances, we use the difference $\sigma _ { 1 } ^ { 2 } - \sigma _ { 2 } ^ { 2 }$ rather than the ratio $\sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 }$ .

Random samples from two normal populations produced the following statistics: s₁ = 3 n₁=30 S₂ = 4 n₂=30 What is the value of the test statistic if we wanted to test the hypothesis that the two populations differ? 0.750 1.333 1.778 0.563

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: A. 0.321. B. 1.009. C. 0.311. D. 0.974.

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

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.

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 $s _ { 1 } ^ { 2 } = 400 \text { and } s _ { 2 } ^ { 2 } = 800$ , and the sample sizes are $n _ { 1 } = 25 \text { and } n _ { 2 } = 25$ . The calculated value of the test statistic will be F = 2.

An investor is considering two types of investment. She is quite satisfied that the expected return on investment 1 is higher than the expected return on investment 2. However, she is quite concerned that the risk associated with investment 1 is higher than that of investment 2. To help make her decision, she randomly selects seven monthly returns on investment 1 and ten monthly returns on investment 2. She finds that the sample variances of investments 1 and 2 are 225 and 118, respectively. Estimate with 95% confidence the ratio of the two population variances, and briefly describe what the interval estimate tells you.

In a random sample of 20 patients who visited the emergency room of hospital 1, a researcher found that the variance of the waiting time (in minutes) was 128.0. In a random sample of 15 patients in the emergency room of hospital 2, the researcher found the variance to be 178.8. Briefly describe what the interval estimate in the previous question tells you.

Random samples from two normal populations produced the following statistics: $n _ { 1 } =$ 25, $s _ { 1 } ^ { 2 } =$ 75. $n _ { 2 } =$ 13, $S _ { 2 } ^ { 2 } =$ 130. Estimate with 95% confidence the ratio of the two population variances.

In a random sample of 20 patients who visited the emergency room of hospital 1, a researcher found that the variance of the waiting time (in minutes) was 128.0. In a random sample of 15 patients in the emergency room of hospital 2, the researcher found the variance to be 178.8. Estimate with 95% confidence the ratio of the two population variances.