Question

A new thermostat has been engineered for the frozen food cases in large supermarkets. Both the old and new thermostats hold temperatures at an average of 25°F. However, it is hoped that the new thermostat might be more dependable in the sense that it will hold temperatures closer to 25°F. One frozen food case was equipped with the new thermostat, and a random sample of 21 temperature readings gave a sample variance of 4.6. Another similar frozen food case was equipped with the old thermostat, and a random sample of 15 temperature readings gave a sample variance of 12.4. Test the claim that the population variance of the new thermostat temperature readings is smaller 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? (Let population 1 refer to data from the old thermostat.)

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: ?₁² = ?₂²; H₁: ?₁² < ?₂

²H₀: ?₁² = ?₂²; H₁: ?₁² > ?₂²   

H₀: ?₁² = ?₂²; H₁: ?₁² ? ?₂²

H₀: ?₁² > ?₂²; H₁: ?₁² = ?₂²

(b) Find the value of the sample F statistic. (Round your answer to two decimal places.)


What are the degrees of freedom?

dfN	=
dfD	=

What assumptions are you making about the original distribution?

The populations follow independent normal distributions.

The populations follow dependent normal distributions. We have random samples from each population.    

The populations follow independent chi-square distributions. We have random samples from each population.

The populations follow independent normal distributions. We have random samples from each population.

(c) Find or estimate the P-value of the sample test statistic. (Round your answer to four decimal places.)


(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the ? = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.    

At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the ? = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is insufficient evidence that the population variance is smaller in the new thermostat temperature readings.

Fail to reject the null hypothesis, there is sufficient evidence that the population variance is smaller in the new thermostat temperature readings.    

Reject the null hypothesis, there is insufficient evidence that the population variance is smaller in the new thermostat temperature readings.

Reject the null hypothesis, there is sufficient evidence that the population variance is smaller in the new thermostat temperature readings.

Answer 1

Let , be the population variance for old thermostat and be the population variance for the new thermostat.

Given :

Sample size for old thermostat = n1 = 15

Sample size for new thermostat = n2 = 21

Sample variance for old thermostat = = 12.4

Sample variance for new  thermostat = = 4.6

(a) The significance level =

Hypothesis :    Vs   

(b) The test statistic under Ho is ,

   Since ,

The degrees of freedom are ,

dfN = 15-1 = 14

dfD = 21-1 = 20

Assumptions :

The populations follow independent normal distributions. We have random samples from each population.

(c) P value =

Therefore, P value = 0.0210

(d) Decision : Here , P value = 0.0210 <

Therefore , At the ? = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

(e) Interpretation :

Reject the null hypothesis. Hence , there is sufficient evidence that the population variance is smaller in the new thermostat temperature readings.