Use the two-standard-deviations F-interval procedure to find the required confidence interval. Assume that independentsamples have been randomly selected from the two populations and that the variable under consideration is normallydistributed on both populations.
-The manager of a juice bottling factory is considering installing a new juice bottling machine which she hopes will reduce the amount of variation in the volumes of juice dispensed into 8-fluid-ounce
Bottles. Random samples of 10 bottles filled by the old machine and 9 bottles filled by the new
Machine yielded the following volumes of juice (in fluid ounces).
Old machine:
$\quad 8.1,8.0,8.0,8.1,8.0,7.9,8.0,7.9,8.2,8.0$
New machine:
$\quad 8.0,8.1,8.0,8.1,7.9,8.0,7.9,8.0,8.1$
Construct a $90 \%$ confidence interval for the ratio, $\sigma _ { 1 } / \sigma _ { 2 }$, where $\sigma _ { 1 }$ is the population standard deviation of the volumes of juice dispensed by the old machine and $\sigma _ { 2 }$ is the population standard deviation of the volumes of juice dispensed by the new machine. (Note: $s _ { 1 } = 0.0919 , s _ { 2 } = 0.0782$ )
A)$0.92$ to $2.57$
B) $0.55$ to $2.02$
C) $0.74$ to $2.30$
D) $0.64$ to $2.11$

Question

Use the two-standard-deviations F-interval procedure to find the required confidence interval. Assume that independentsamples have been randomly selected from the two populations and that the variable under consideration is normallydistributed on both populations.
-The manager of a juice bottling factory is considering installing a new juice bottling machine which she hopes will reduce the amount of variation in the volumes of juice dispensed into 8-fluid-ounce
Bottles. Random samples of 10 bottles filled by the old machine and 9 bottles filled by the new
Machine yielded the following volumes of juice (in fluid ounces). 
Old machine: 
$\quad 8.1,8.0,8.0,8.1,8.0,7.9,8.0,7.9,8.2,8.0$
New machine:
$\quad 8.0,8.1,8.0,8.1,7.9,8.0,7.9,8.0,8.1$
Construct a $90 \%$ confidence interval for the ratio, $\sigma _ { 1 } / \sigma _ { 2 }$, where $\sigma _ { 1 }$ is the population standard deviation of the volumes of juice dispensed by the old machine and $\sigma _ { 2 }$ is the population standard deviation of the volumes of juice dispensed by the new machine. (Note: $s _ { 1 } = 0.0919 , s _ { 2 } = 0.0782$ )  
A)$0.92$ to $2.57$
B) $0.55$ to $2.02$
C) $0.74$ to $2.30$
D) $0.64$ to $2.11$

Quizplus · Accepted Answer

The confidence interval for the ratio of two variances (or standard deviations) can be found using the F-distribution. Given sample standard deviations $s_1 = 0.0919$ and $s_2 = 0.0782$, and sample sizes $n_1 = 10$ and $n_2 = 9$, we use the formula for the confidence interval for the ratio of variances $\sigma_1^2 / \sigma_2^2$. The ratio of the sample variances $s_1^2 / s_2^2$ serves as the point estimate. The confidence interval is then calculated using the F-distribution's critical values for a given confidence level (90% in this case). The correct interval must account for the degrees of freedom for each sample ($n_1 - 1$ and $n_2 - 1$) and the properties of the F-distribution. The provided intervals are for the ratio of standard deviations, not variances, so the calculation involves squaring the sample standard deviations if working directly with variances. The correct choice, given the calculation method and the provided sample statistics, reflects the appropriate F-distribution critical values and the calculation for the ratio of standard deviations.

Use the Two-Standard-Deviations F-Interval Procedure to Find the Required Confidence 8.1,8.0,8.0,8.1,8.0,7.9,8.0,7.9,8.2,8.0\quad 8.1,8.0,8.0,8.1,8.0,7.9,8.0,7.9,8.2,8.08.1,8.0,8.0,8.1,8.0,7.9,8.0,7.9,8.2,8.0

Use the Two-Standard-Deviations F-Interval Procedure to Find the Required Confidence $\quad 8.1,8.0,8.0,8.1,8.0,7.9,8.0,7.9,8.2,8.0$