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 4.7. Another similar frozen food case was equipped with the old thermostat, and a random sample of 14 temperature readings gave a sample variance of 12.2. 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? (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 ≠ σ₂²H₀: σ₁² = σ₂²; H₁: σ₁² > σ₂²    H₀: σ₁² > σ₂²; H₁: σ₁² = σ₂²H₀: 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?

dfN	=
dfD	=

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

p-value > 0.1000.050 < p-value < 0.100    0.025 < p-value < 0.0500.010 < p-value < 0.0250.001 < p-value < 0.010p-value < 0.001

(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 statistically significant.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.

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

Reject the null hypothesis, there is sufficient evidence that the population variance is larger in the old thermostat temperature readings.Fail to reject the null hypothesis, there is sufficient evidence that the population variance is larger in the old thermostat temperature readings.    Reject the null hypothesis, there is insufficient evidence that the population variance is larger in the old thermostat temperature readings.Fail to reject the null hypothesis, there is insufficient evidence that the population variance is larger in the old thermostat temperature readings.

Answer 1

a).i) level of significance () =0.05

ii).the null and alternate hypotheses:-

b).given data:-

sample variance of old thermostat temperature readings() = 12.2

sample size of old thermostat temperature readings () = 14

sample variance of new thermostat temperature readings() = 4.7

sample size of new thermostat temperature readings () = 26

i). the value of the sample F statistic:-

ii).degrees of freedom:-

numerator
denominator

iii).assumptions i am making about the original distribution:-

  We have random samples from each population.  

  The populations follow independent normal distributions.

c). p value = 0.0196

( using p value calculator for df= 13 and 25, F =2.60)

estimation of  the P-value of the sample test statistic:-

  0.010 < p-value < 0.0250

d). here, p value = 0.0196 <0.05 ,

Based on the answers in parts (a) to (c), i will :-

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

e)conclusion in the context of the application:-

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

***If you have any doubt regarding the problem please write it in the comment section.if you are satisfied please give me a LIKE if possible...