Question

Let *x* = age in years of a rural Quebec woman at the
time of her first marriage. In the year 1941, the population
variance of *x* was approximately *σ*^{2} =
5.1. Suppose a recent study of age at first marriage for a random
sample of 31 women in rural Quebec gave a sample variance
*s*^{2} = 2.5. Use a 5% level of significance to
test the claim that the current variance is less than 5.1. Find a
90% confidence interval for the population variance.

(a) What is the level of significance?

State the null and alternate hypotheses.

*H _{o}*:

(b) Find the value of the chi-square statistic for the sample.
(Round your answer to two decimal places.)

What are the degrees of freedom?

What assumptions are you making about the original
distribution?

We assume a exponential population distribution.We assume a uniform population distribution. We assume a normal population distribution.We assume a binomial population distribution.

(c) Find or estimate the *P*-value of the sample test
statistic.

*P*-value > 0.1000.050 < *P*-value <
0.100 0.025 < *P*-value <
0.0500.010 < *P*-value < 0.0250.005 <
*P*-value < 0.010*P*-value < 0.005

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

Since the *P*-value > *α*, we fail to reject
the null hypothesis.Since the *P*-value > *α*, we
reject the null hypothesis. Since the
*P*-value ≤ *α*, we reject the null hypothesis.Since
the *P*-value ≤ *α*, we fail to reject the null
hypothesis.

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

At the 5% level of significance, there is insufficient evidence to conclude that the variance of age at first marriage is less than 5.1.At the 5% level of significance, there is sufficient evidence to conclude that the that the variance of age at first marriage is less than 5.1.

(f) Find the requested confidence interval for the population
variance. (Round your answers to two decimal places.)

lower limit | |

upper limit |

Interpret the results in the context of the application.

We are 90% confident that *σ*^{2} lies within
this interval.We are 90% confident that *σ*^{2} lies
below this interval. We are 90% confident
that *σ*^{2} lies outside this interval.We are 90%
confident that *σ*^{2} lies above this interval.

Answer #1

Part a)

α = 0.05

*H _{o}*:

Part b)

Test Statistic :-

χ^{2} = ( ( 31-1 ) * 2.5 ) / 5.1

**χ ^{2} = 14.72**

Degree of freedom = n - 1 = 31 - 1 = 30

We assume a normal population distribution.

Part c)

P value = P ( χ^{2} > 14.7059 ) = 0.0087

0.005 < *P*-value < 0.010

Decision based on P value

P value = P ( χ^2 > 14.7059 )

P value = 0.0087

Reject null hypothesis if P value < α = 0.05

Since P value = 0.0087 < 0.05, hence we reject the null
hypothesis

Conclusion :- We Reject H0

Part d)

Since the *P*-value ≤ *α*, we reject the null
hypothesis.

Part e)

At the 5% level of significance, there is sufficient evidence to conclude that the that the variance of age at first marriage is less than 5.1.

Part f)

((n-1)S^{2} / χ^{2} (0.05/2)) < σ^{2}
< ((n-1)S^{2} / χ^{2} (1 - 0.05/2) )

(( 31-1 ) 2.5 / χ^{2} (0.05/2) ) < σ^{2} <
((31-1)2.5 / χ^{2} (1 - 0.05/2) )χ^2 (0.1/2) = 43.773

χ^{2} (0.1/2) = 43.773

χ^{2} (1 - 0.1/2) ) = 18.4927

Lower Limit = (( 31-1 ) 2.5^{2} / χ^{2} (0.1/2) ) =
1.7134

Upper Limit = (( 31-1 ) 2.5^{2} / χ^{2} (0.1/2) ) =
4.0557

90% Confidence interval is ( 1.7134 , 4.0557 )

**( 1.71 < σ ^{2} < 4.06 )**

We are 90% confident that *σ*^{2} lies within
this interval.

Let x = age in years of a rural Quebec woman at the
time of her first marriage. In the year 1941, the population
variance of x was approximately σ2 =
5.1. Suppose a recent study of age at first marriage for a random
sample of 51 women in rural Quebec gave a sample variance
s2 = 3.4. Use a 5% level of significance to
test the claim that the current variance is less than 5.1. Find a
90% confidence...

Let x = age in years of a rural Quebec woman at the
time of her first marriage. In the year 1941, the population
variance of x was approximately σ2 =
5.1. Suppose a recent study of age at first marriage for a random
sample of 41 women in rural Quebec gave a sample variance
s2 = 3.1. Use a 5% level of significance to
test the claim that the current variance is less than 5.1. Find a
90% confidence...

Let x = age in years of a rural Quebec woman at the
time of her first marriage. In the year 1941, the population
variance of x was approximately σ2 =
5.1. Suppose a recent study of age at first marriage for a random
sample of 51 women in rural Quebec gave a sample variance
s2 = 2.8. Use a 5% level of significance to
test the claim that the current variance is less than 5.1. Find a
90% confidence...

Let x = age in years of a rural Quebec woman at the
time of her first marriage. In the year 1941, the population
variance of x was approximately σ2 =
5.1. Suppose a recent study of age at first marriage for a random
sample of 31 women in rural Quebec gave a sample variance
s2 = 2.4. Use a 5% level of significance to
test the claim that the current variance is less than 5.1. Find a
90% confidence...

Let x = age in years of a rural Quebec woman at the
time of her first marriage. In the year 1941, the population
variance of x was approximately σ2 =
5.1. Suppose a recent study of age at first marriage for a random
sample of 51 women in rural Quebec gave a sample variance
s2 = 3.5. Use a 5% level of significance to
test the claim that the current variance is less than 5.1. Find a
90% confidence...

Let x = age in years of a rural Quebec woman at the
time of her first marriage. In the year 1941, the population
variance of x was approximately σ2 =
5.1. Suppose a recent study of age at first marriage for a random
sample of 51 women in rural Quebec gave a sample variance
s2 = 2.5. Use a 5% level of significance to
test the claim that the current variance is less than 5.1. Find a
90% confidence...

Let x = age in years of a rural Quebec woman at the
time of her first marriage. In the year 1941, the population
variance of x was approximately σ2 =
5.1. Suppose a recent study of age at first marriage for a random
sample of 51 women in rural Quebec gave a sample variance
s2 = 3.1. Use a 5% level of significance to
test the claim that the current variance is less than 5.1. Find a
90% confidence...

Let x = age in years of a rural Quebec woman at the time of her
first marriage. In the year 1941, the population variance of x was
approximately σ2 = 5.1. Suppose a recent study of age at first
marriage for a random sample of 51 women in rural Quebec gave a
sample variance s2 = 3.1. Use a 5% level of significance to test
the claim that the current variance is less than 5.1. Find a 90%
confidence...

Let x = age in years of a rural Quebec woman at the
time of her first marriage. In the year 1941, the population
variance of x was approximately σ2 =
5.1. Suppose a recent study of age at first marriage for a random
sample of 41 women in rural Quebec gave a sample variance
s2 = 2.5. Use a 5% level of significance to
test the claim that the current variance is less than 5.1. Find a
90% confidence...

Let x = age in years of a rural Quebec woman at the time of her
first marriage. In the year 1941, the population variance of x was
approximately σ2 = 5.1. Suppose a recent study of age at first
marriage for a random sample of 41 women in rural Quebec gave a
sample variance s2 = 2.4. Use a 5% level of significance to test
the claim that the current variance is less than 5.1. Find a 90%
confidence...

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