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 5.1. Another similar
frozen food case was equipped with the old thermostat, and a random
sample of 14 temperature readings gave a sample variance of 12.9.
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.

*H*_{0}: *σ*_{1}^{2} =
*σ*_{2}^{2}; *H*_{1}:
*σ*_{1}^{2} <
*σ*_{2}^{2}

*H*_{0}: *σ*_{1}^{2} =
*σ*_{2}^{2}; *H*_{1}:
*σ*_{1}^{2} >
*σ*_{2}^{2}

*H*_{0}: *σ*_{1}^{2} >
*σ*_{2}^{2}; *H*_{1}:
*σ*_{1}^{2} =
*σ*_{2}^{2}

*H*_{0}: *σ*_{1}^{2} =
*σ*_{2}^{2}; *H*_{1}:
*σ*_{1}^{2} ≠
*σ*_{2}^{2}

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

What are the degrees of freedom?

df_{N} |
= |

df_{D} |
= |

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.)

*p*-value > 0.1000

.050 < *p*-value <
0.100

0.025 < *p*-value < 0.0500

.010 < *p*-value < 0.0250

.001 < *p*-value < 0.010

*p*-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 fail to 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 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 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 sufficient 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)

5%

*H*_{0}: *σ*_{1}^{2} =
*σ*_{2}^{2}; *H*_{1}:
*σ*_{1}^{2} >
*σ*_{2}^{2}

(b)

dfN = 20

dfD = 13

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

F = s1^2 / s2^2 = 5.1/12.9 = 0.40

(c)

p- value = 0.9698

*p*-value > 0.1000

(d)

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

(e)

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

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 5.3. Another similar
frozen food case was...

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 4.4. Another similar
frozen food case was...

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...

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 4.7. Another similar
frozen food case was...

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...

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...

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...

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 4.6. Another similar
frozen food case was...

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