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A new thermostat has been engineered for the frozen food cases in large supermarkets. Both the old and the new thermostats hold temperatures at an average of \(25^{\circ} \mathrm{F}\). However, it is hoped that the new thermostat might be more dependable in the sense that it will hold temperatures closer to \(25^{\circ} \mathrm{F}\). One frozen food case was equipped with the new thermostat, and a random sample of 21 temperature readings gave a sample variance of \(5.1 .\) Another, similar frozen food case was equipped with the old thermostat, and a random sample of 16 temperature readings gave a sample variance of \(12.8 .\) Test the claim that the population variance of the old thermostat temperature readings is larger than that for the new thermostat. Use a \(5 \%\) level of significance. How could your test conclusion relate to the question regarding the dependability of the temperature readings?

Step by step solution

01

Identify the Hypotheses

We need to perform a hypothesis test to determine if the population variance of the old thermostat is greater than that of the new thermostat. The null hypothesis \(H_0\) is that the variances are equal, i.e., \(\sigma^2_{old} \leq \sigma^2_{new}\). The alternative hypothesis \(H_a\) is that the variance of the old thermostat is greater than that of the new one, i.e., \(\sigma^2_{old} > \sigma^2_{new}\). This is a right-tailed test.

02

Determine the Test Statistic

Since the sample sizes are different and we are comparing variances, we will use the F-test for equality of variances. The test statistic is given by \( F = \frac{s^2_{old}}{s^2_{new}} \), where \( s^2_{old} = 12.8 \) and \( s^2_{new} = 5.1 \). Calculate \( F \) as follows: \[F = \frac{12.8}{5.1} \approx 2.5098\]

03

Find the Critical Value

To find the critical value for the test, we use the F-distribution table with degrees of freedom \( df_1 = n_{old} - 1 = 15 \) and \( df_2 = n_{new} - 1 = 20 \) at a significance level of \( \alpha = 0.05 \). From the F-distribution table, the critical value for \( F(15, 20, 0.05) \) is approximately 2.54.

04

Make a Decision

Compare the calculated F-statistic with the critical value. Our calculated F-statistic is approximately 2.5098, and the critical value is 2.54. Since 2.5098 is less than 2.54, we fail to reject the null hypothesis.

05

Interpret the Results

Since we failed to reject the null hypothesis, there is not enough statistical evidence at the 5% significance level to support the claim that the variance of the old thermostat is greater than the variance of the new thermostat. This suggests that we do not have enough evidence to conclude that the new thermostat is more dependable in terms of variance reduction compared to the old thermostat.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

F-test in Hypothesis Testing

When comparing variances between two groups, the F-test is a powerful statistical method to use. It helps us decide whether differences in sample variances suggest a real variance difference in the populations they come from. In our exercise, we needed to check if the variance of the old thermostat's temperature readings was indeed larger than that of the new thermostat. To do this, we applied the F-test.

The F-test works by comparing the ratio of two variances. If we call the variance of the first sample as \(s_1^2\) and the second as \(s_2^2\), the F-statistic is determined by \[F = \frac{s_1^2}{s_2^2}.\]

Here, the larger variance should be placed in the numerator to align with the test's direction (right-tailed, left-tailed, or two-tailed). By examining this ratio against critical values from the F-distribution table, we can assess whether any observed differences are statistically significant.

Understanding Variance Comparison

Variance comparison looks at the degree of spread in data points. In a practical context, this tells us how consistent each thermostat maintains a specific temperature.

For the exercise, the new thermostat had a sample variance of 5.1, while the old thermostat's variance was 12.8. Variance measures how far each data point in your set extends from the overall mean. A lower variance indicates that data points tend to be closer to the mean, which suggests higher consistency or dependability.

• A smaller variance in thermostat implies it closely maintains the target temperature with minor fluctuations.
• A larger variance suggests more deviation in temperature readings from the expected mean.

In hypothesis testing, comparing variances using tools like the F-test allows us to statistically assess whether changes in spread are meaningful or could just be due to random sample variations.

Role of Significance Level in Hypothesis Testing

In hypothesis testing, the significance level (also known as \( \alpha \)) is critical for decision-making. It represents the threshold for deciding whether to reject the null hypothesis.

In our exercise context, the significance level was set at 0.05 or 5%. This means we accept a 5% risk of concluding that the old thermostat's variance is greater when it isn't.

• A lower \( \alpha \) value implies stricter criteria for accepting evidence in favor of the alternative hypothesis.
• A higher \( \alpha \) allows for more leniency, potentially increasing the chance of a Type I error (incorrectly rejecting the null hypothesis).

Therefore, a significance level of 0.05 is commonly used as it balances the risk of errors while still being open to detect genuine differences. In situations like the thermostat study, this level guides us in rightly concluding whether the new thermostat's consistency is justifiably better in practice.