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 26 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 19 temperature readings gave a sample variance of 12.3. Test the claim that the population variance of the new thermostat temperature readings is smaller than that for the old 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: σ12 = σ22; H1: σ12 < σ22

H0: σ12 = σ22; H1: σ12 ≠ σ22

H0: σ12 > σ22; H1: σ12 = σ22

H0: σ12 = σ22; H1: σ12 > σ22

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

What are the degrees of freedom?

What assumptions are you making about the original distribution?

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.

The populations follow independent normal distributions.

The populations follow dependent 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.)

p-value > 0.100

0.050 < p-value < 0.100

0.025 < p-value < 0.050

0.010 < p-value < 0.025

0.001 < p-value < 0.010

p-value < 0.001 Changed

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

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

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

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

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

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

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 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 sufficient evidence that the population variance is smaller in the new thermostat temperature readings.

Answer 1

Claim: The population variance of the new thermostat temperature readings is smaller than that for the old thermostat.

a) Level of significance = 0.05

The null and alternative hypothesis is

H0: σ12 = σ22; H1: σ12 < σ22

b) Test statistic is

F = largest sample variance / Smallest sample variances

F = 12.3 / 5.1 = 2.41

Degrees of freedom => n1 - 1 , n2 - 1 => 26 - 1 , 19 - 1 => 25 , 18

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

c) P-value is lies between

0.025 < p-value < 0.050

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

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