Question

Let x represent the number of mountain climbers killed each year. The long-term variance of x is approximately σ² = 136.2. Suppose that for the past 6 years, the variance has been s² = 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.

(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?

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

lower limit
upper limit

Answer 1

Chi-Square Test for one population variance

The Chi-Square test for one population variance is used to test whether the sample variance is greater or less than 0.01. The test is performed in the following steps,

Step 1: Hypothesis

The null and alternative hypotheses are,

This is a left-tailed test

Step 2: Decision rule

The critical value for the chi-square statistic is obtained from chi-square distribution table for significance level = 0.01 and degree of freedom = n -1 = 6 - 1 = 5.

Since this is left tailed test, reject the null hypothesis if,

Step 3: Test statistic

The Chi-Squared statistic is obtained using the following formula,

Step 4: Conclusion

Since the chi-square statistic is greater than the lower critical value, the null hypothesis is not rejected. hence there is not sufficient evidence to conclude that the variance for the number of mountain-climber deaths is less than 136.2.

b)

From the above hypothesis test,

the chi-square statistic = 3.9317

the degrees of freedom = 5

c)

Answer:

Lower limit = 48.3718

Upper limit = 497.4912

Explanation:

The 90% confidence interval for the variance

The 90% confidence interval for variance is obtained using the following formula,

where

Sample size = 5

sample variance = 107.1

The critical value for chi-square is obtained from chi-square distribution table table, for significance level = 0.10 degree of freedom = n - 1 = 31

Left tailed

Right tailed

now,