Question

A comparison of the precisions of two machines developed for extracting juice from oranges is to be made using the following data: $$ \begin{array}{ll} \text { Machine A } & \text { Machine B } \\ \hline s^{2}=3.1 \text { ounces }^{2} & s^{2}=1.4 \text { ounces }^{2} \\ n=25 & n=25 \end{array} $$ a. Is there sufficient evidence to indicate that there is a difference in the precision of the two machines at the \(5 \%\) level of significance? b. Find a \(95 \%\) confidence interval for the ratio of the two population variances. Does this interval confirm your conclusion from part a? Explain.

Short answer

Based on the calculations and analysis, we can conclude that there is sufficient evidence to indicate a difference in the precision of the two machines at a 5% level of significance. Moreover, the 95% confidence interval for the ratio of the population variances, which ranges from 0.896 to 4.25, confirms this conclusion as the interval does not contain 1, indicating a significant difference in the precision of the machines.

Step-by-step solution

Compute the F-statistic

Since we are comparing the variances, we need to calculate the F-statistic for the sample variances (\(s^{2}_{A}\) and \(s^{2}_{B}\)) of Machine A and Machine B. $$ F = \frac{s^{2}_{A}}{s^{2}_{B}} = \frac{3.1}{1.4} \approx 2.21 $$

Compute the critical F-value

We are given a 5% level of significance, so we need to find the critical F-value for a 0.05 significance level (with \(n_A - 1 = 24\) and \(n_B - 1 = 24\) degrees of freedom) using an F-distribution table or software. Based on an F-distribution table, we find that the critical F-value for \(\alpha=0.05\) and with \((24, 24)\) degrees of freedom is approximately F(0.975, 24, 24) = 1.92.

Compare the F-statistic with the critical F-value

Now we need to compare the calculated F-statistic with the critical F-value to determine if there is a significant difference in the precisions of the two machines. Since the F-statistic (2.21) is greater than the critical F-value (1.92), we can reject the null hypothesis and conclude that there is sufficient evidence to indicate a difference in the precision of the two machines at a 5% level of significance. #b. Find a 95% confidence interval for the ratio of the population variances. Does this interval confirm your conclusion from part a? Explain.#

Compute the confidence interval for the ratio of population variances

A 95% confidence interval for the ratio of population variances is calculated with the following formula: $$ \frac{s^{2}_{A}}{s^{2}_{B}} \times \frac{1}{F_{\alpha/2, n_A - 1, n_B - 1}} \leq \frac{\sigma^{2}_{A}}{\sigma^{2}_{B}} \leq \frac{s^{2}_{A}}{s^{2}_{B}} \times F_{\alpha/2, n_A - 1, n_B - 1} $$ We know that our F-value is approximately 2.21 (\(\frac{s^{2}_{A}}{s^{2}_{B}}\)), and from the F-distribution table, we find that \(F_{0.025, 24, 24} \approx 2.469\) and \(F_{0.975, 24, 24} \approx 1.92\). Hence, our 95% confidence interval will be: $$ 2.21 \times \frac{1}{2.469} \leq \frac{\sigma^{2}_{A}}{\sigma^{2}_{B}} \leq 2.21 \times 1.92 $$ $$ 0.896 \leq \frac{\sigma^{2}_{A}}{\sigma^{2}_{B}} \leq 4.25 $$

Analyze the confidence interval

The 95% confidence interval calculated in step 4 suggests that the true ratio of population variances lies between 0.896 and 4.25. Since the interval doesn't contain 1 (which would indicate no difference between the variances), this result confirms our conclusion from part a, indicating that there is a significant difference in the precision of the two machines.

Key concepts

F-distribution

The F-distribution is a crucial concept when comparing variances in statistics. It takes its name from the statistician Sir Ronald Fisher. This distribution is used in hypothesis testing to determine if there are significant differences between the variances of two or more groups. It is especially relevant when analyzing data from experiments or machinery like in the orange juice extraction example.

The F-distribution is characterized by two sets of degrees of freedom: one for the numerator and one for the denominator. In our case, we are looking at 24 degrees of freedom for both machines since both samples had 25 observations. These degrees of freedom impact the shape of the F-distribution.

• The F-distribution is not symmetrical; it is positively skewed and only takes positive values.
• As the degrees of freedom increase, the distribution becomes less skewed and more symmetrical.
• This distribution helps us understand the ratio of two variances and make inferences about them.

To use the F-distribution effectively, it is essential to compare the calculated F-statistic against critical values from an F-distribution table or software to determine if the observed variance is significantly different.

Confidence Interval

A confidence interval offers a range of values for an unknown parameter (like the ratio of population variances), which we estimate using our sample data. It is an interval estimate that, with a certain confidence level, contains the true parameter value. Confidence intervals are quite insightful because they provide not just an estimate but a range within which we believe the true value lies.

In our example, we computed a 95% confidence interval for the ratio of the population variances of machine precisions. This means we are 95% confident that this interval contains the true ratio.

• To compute the interval for the ratio of population variances, we use the F-statistic and critical F-values from the F-distribution.
• The lower bound is calculated using \( \frac{s^{2}_{A}}{s^{2}_{B}} \times \frac{1}{F_{\alpha/2, n_A - 1, n_B - 1}} \), while the upper bound is \( \frac{s^{2}_{A}}{s^{2}_{B}} \times F_{\alpha/2, n_A - 1, n_B - 1} \).
• This interval allows us to determine if the variance of one machine is significantly different from another.

In practice, if the interval includes the value 1, it suggests that there might not be a significant difference between the variances. But if 1 is not in the interval, as in our example, it indicates a potentially significant difference.

Population Variance

Population variance is a measure of how data points in a population are spread out. It is an important concept in statistics, especially when determining the reliability and precision of measurement instruments like juice extraction machines. In essence, it indicates the "spread" or "dispersion" of all the values in a population.

• Variances are calculated by taking the average of squared deviations from the mean.
• Larger variances indicate that numbers are more spread out from the mean.
• When comparing two machines, if one has a significantly larger variance, it may be less precise in its measurements.

Population variance can be deduced from sample variance when the sample size is representative of the population. In our exercise, using sample variances, we make inferences about population variances to assess machine precision.

Level of Significance

The level of significance is a key concept in hypothesis testing. It represents the probability threshold below which we reject the null hypothesis. Often denoted by \( \alpha \), it indicates the risk we are willing to take of rejecting a true null hypothesis.

In the juice machine example, the level of significance is set at 5% (or 0.05). This means we are willing to accept a 5% chance of concluding there is a difference in machine precision when, in reality, there might not be any difference.

• A lower level of significance means a more stringent criterion for rejecting the null hypothesis.
• Choosing a level of significance involves a trade-off between type I error (rejecting a true null hypothesis) and type II error (failing to reject a false null hypothesis).
• Common significance levels are 0.05, 0.01, and 0.10.

Once an \( \alpha \) is chosen, it guides the critical value selection from our F-distribution table, which we compare against our F-statistic to make decisions in our hypothesis test.