Hypothesis Testing Question

[CFAgrind]

Hi Guys,

I have been studying all day and have run into a problem that I can’t seem to figure out (maybe because I’m mentally exhausted today). I’m hoping someone can help me out here:

Question:

A study was conducted to determine whether the standard deviation of monthly maintenance costs of a Pepper III aircraft is $300. A sample of 30 Pepper III’s had a mean monthly maintenance cost of $3025 and a standard deviation of $325. Using a 5% level of significance, which of the following is the most appropriate conclusion regarding difference between the hypothesized value of the populatin variance and the sample variance?

A. The population and sample variances are significantly different.

B. The population and sample variances are not significantly different.

C. There are no tests that may be used to test variance differences in small samples.

Answer:

I understand how to get the test statistic (which is 34.035).

What I do not understand is how do I find the critical chi-square values (which the solution manual says are 16.047 on the left and 45.722 on the right). If anyone can help me out I would really appreciate it!

Thanks in advance!

[breadmaker]

You have 29 degrees of freedom = 30 - 1.

If you look at chi squared table, you need the values that correspond to 2.5% and 97.5% cumulative limits with 29 d.f. These values are your 16.047 and 45.722, respectively. In essence, you are calculating a 95% confidence interval. Your test statistic falls within this interval, so you do not reject the hypothesis that the population standard deviation is 300.

Hope this helps!

[breadmaker]

You have 29 degrees of freedom = 30 - 1. You lose a degree of freedom using the sample mean in place of the population mean.

If you look at a chi squared table, you need the values that correspond to the 2.5% and 97.5% cumulative limits with 29 d.f. These values are your 16.047 and 45.722, respectively. In essence, you are calculating a 95% confidence interval. Your test statistic falls within this interval, so you do not reject the hypothesis that the population standard deviation is 300.

Hope this helps!

[breadmaker]

Oops! The website left me hanging, so I didn’t think my post went through!

[CFAgrind]

@breadmaker

You have 29 degrees of freedom = 30 - 1.

If you look at chi squared table, you need the values that correspond to 2.5% and 97.5% cumulative limits with 29 d.f. These values are your 16.047 and 45.722, respectively. In essence, you are calculating a 95% confidence interval. Your test statistic falls within this interval, so you do not reject the hypothesis that the population standard deviation is 300.

Hope this helps!

That’s a great help! That you so much.

We are not required to know the chi-squared tables on the exam though, correct?

[CFAgrind]

@breadmaker

You have 29 degrees of freedom = 30 - 1.

If you look at chi squared table, you need the values that correspond to 2.5% and 97.5% cumulative limits with 29 d.f. These values are your 16.047 and 45.722, respectively. In essence, you are calculating a 95% confidence interval. Your test statistic falls within this interval, so you do not reject the hypothesis that the population standard deviation is 300.

Hope this helps!

That’s a great help! That you so much.

We are not required to know the chi-squared tables on the exam though, correct?

[breadmaker]

They would have to give you the tables: the probability density function depends on the number of degrees of freedom and is rather nasty to integrate.

If I were on the exam committee, what I would do is give you the values for d.f.=28,29,30 and cumulative limits of 1%, 2.5%, 5%, 95%, 97.5%, and 99%. I would see if you could pick out the correct values.