Answer to Question #263102 in Statistics and Probability for John

Question #263102

Test Statistic for σ2 is given by: χ2 = (n −1)s 2 σ2 ; which is Chi-squared distribbuted with ν = n −1 degrees of freedom when the population random variable is normally distributed with variance equal to σ2 3 (a) A random sample of n = 100 observations was drawn from a normal population. The sample variance was calculated to be s 2 = 220. Test with α = 0.05 to determine whether we can infer that the population variance differs from 300. (b) Repeat part (a) changing the sample size to n = 50 (c) What is the effect of decreassing the sample size?


1
Expert's answer
2021-11-09T16:32:52-0500

From the information we are given,

"n=100,\\space s^2=220"

a.

The hypotheses to be tested are,

"H_0:\\sigma^2=300,\\space against\\space H_1:\\sigma\\not=300"

As given above, the test statistic is,

"\\chi^2=(n-1)*s^2\/\\sigma^2=99*220\/300=72.6"

"\\chi^2" is compared with a chi-squared table value at "\\alpha=5\\%" with "v=n-1=100-1=99" degrees of freedom.

The table value is given as,

"\\chi^2_{\\alpha\/2,v}=\\chi^2_{0.05\/2,99}=\\chi^2_{0.025,99}=128.422" and the null hypothesis is rejected if "\\chi^2\\gt \\chi^2_{0.025,99}"

Since "\\chi^2=72.6\\lt\\chi^2_{0.025,99}=128.422," we fail to reject the null hypothesis and conclude that there is insufficient evidence to show that the population variance differs from 300 at 5% level of significance.


b.

when "n=50",

The test statistic is given as,

"\\chi^2=(50-1)*220\/300=35.93333" which is compared with the table value at "\\alpha=0.05" with "v=n-1=50-1=49" degrees of freedom.

The table value is given as,

"\\chi^2_{0.05\/2,49}=\\chi^2_{0.025,49}=70.22241" and the null hypothesis is rejected if "\\chi^2\\gt\\chi^2_{0.05,49}"

Since "\\chi^2=35.93333\\lt\\chi^2_{0.025,49}=70.22241", we fail to reject the null hypothesis and conclude that there is insufficient evidence to show that the population variance differs from 300 at 5% level of significance.


c.

Clearly, decreasing the sample size decreases the value of the test statistic as well as the table value.



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