Question

Let *x* represent the number of mountain climbers killed
each year. The long-term variance of *x* is approximately
*σ*^{2} = 136.2. Suppose that for the past 11 years,
the variance has been *s*^{2} = 107.1. Use a 1%
level of significance to test the claim that the recent variance
for number of mountain-climber deaths is less than 136.2. Find a
90% confidence interval for the population variance.

(a) What is the level of significance?

State the null and alternate hypotheses.

*H _{o}*:

*H _{o}*:

*H _{o}*:

*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 binomial population distribution.

We assume a uniform population distribution.

We assume a normal population distribution.

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

*P*-value > 0.100

0.050 < *P*-value < 0.100

0.025 < *P*-value < 0.050

0.010 < *P*-value < 0.025

0.005 < *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 1% level of significance, there is insufficient evidence to conclude that the variance for number of mountain climber deaths is less than 136.2

At the 1% level of significance, there is sufficient evidence to conclude that the variance for number of mountain climber deaths is less than 136.2

(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 outside
this interval.

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

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

Answer #1

a) level of significance =0.01

*H _{o}*:

b)

value of the chi-square statistic =(11-1)*107.1/136.2=7.86

degrees of freedom =n-1=10

We assume a normal population distribution

c)

*P*-value > 0.100

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

e)

At the 1% level of significance, there is insufficient evidence to conclude that the variance for number of mountain climber deaths is less than 136.2

f)

lower limit =58.50

upper limit =271.83

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

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