Inferences involving 2 populations

prettylittlepixels

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16 "growth" stocks (G) and 10 "income" stocks (I) are randomly selected, and their rates of return for the last year are measured. We are interested in the variability of the rates of return for the two types of stock, not the rates of return themselves.

Growth (G): sample size n = 16 ; variance = 34.7
Income (I): sample size n = 10 ; variance = 114.9


1.1
Test the hypothesis that the standard deviations of these two populations are equal at the 5% level of significance. What is the null hypothesis?

sigma^2(I) / sigma^2(G) <or= 1


1.2

Test the hypothesis that the standard deviations of these two populations are equal at the 5% level of significance. Placing the larger variance in the numerator, as the textbook suggests, what is the critical value for this hypothesis test?
2.59

1.3
Test the hypothesis that the standard deviations of these two populations are equal at the 5% level of significance. What is your decision?
The standard deviation for growth stocks is less than the standard deviation for income stocks at the 5% level of significance.
1.4
Create a 90% confidence interval estimate of the true ratio of the standard deviations, using the income stocks (I) in the numerator. Use the confidence interval Formula F from Diana's handout "Econ 262 Formulas" and remember that you are calculating the ratio of STANDARD DEVIATIONS, not the ratio of variances.

What is the lower confidence limit and what is the upper confidence limit??
I calculated .703 and emailed my teacher that I wasn’t sure what to add and subtract this number from, but he said it was wrong. I’m not sure what I’m doing wrong. He said to use the formula below:

F test of two variances or standard deviations
F* = s12/s22 where dfn = n1 – 1 and dfd = n2 – 1
For two-tailed tests, using the larger sample variance in the numerator will eliminate the necessity of calculating the left-tailed critical value.

For one-tailed left-tailed tests, using the larger sample variance in the numerator will turn the test into a one-tailed right-tailed test.

For left-tailed critical values, F(dfn, dfd, 1-α/2) = 1 / F(dfd, dfn, α/2)

(1 – α) Confidence Interval for σ12/σ22 is

s12/s22 to (s12/s22) F(df2, df1, α/2)
F(df1, df2, α/2)

1.5
Does the confidence interval you calculated give you EXACTLY the same information as the hypothesis test above?
A. No. The information is not the same and the decisions are not the same.
B. While the information is the same, the decisions are not the same.
C. Yes. The information is the same and the decisions are the same.
D. While the decisions are the same, the information is not exactly the same.

Any help on this would be greatly appreciated. I have worked into the problem this far and I am lost on 1.4. If you could check over the other components of the problem to see if I did them right, that would be great as well. I’m really struggling in Statistics lately. Thanks!
 
Last edited:
16 "growth" stocks (G) and 10 "income" stocks (I) are randomly selected, and their rates of return for the last year are measured. We are interested in the variability of the rates of return for the two types of stock, not the rates of return themselves.

Growth (G): sample size n = 16 ; variance = 34.7
Income (I): sample size n = 10 ; variance = 114.9


1.1
Test the hypothesis that the standard deviations of these two populations are equal at the 5% level of significance. What is the null hypothesis?

sigma^2(I) / sigma^2(G) <or= 1

So what was your conclusion - can you reject or "cannot reject" your hypothesis?
1.2

Test the hypothesis that the standard deviations of these two populations are equal at the 5% level of significance. Placing the larger variance in the numerator, as the textbook suggests, what is the critical value for this hypothesis test?
2.59

1.3
Test the hypothesis that the standard deviations of these two populations are equal at the 5% level of significance. What is your decision?
The standard deviation for growth stocks is less than the standard deviation for income stocks at the 5% level of significance.
1.4
Create a 90% confidence interval estimate of the true ratio of the standard deviations, using the income stocks (I) in the numerator. Use the confidence interval Formula F from Diana's handout "Econ 262 Formulas" and remember that you are calculating the ratio of STANDARD DEVIATIONS, not the ratio of variances.

What is the lower confidence limit and what is the upper confidence limit??
I calculated .703 and emailed my teacher that I wasn’t sure what to add and subtract this number from, but he said it was wrong. I’m not sure what I’m doing wrong. He said to use the formula below: (Did you?)

F test of two variances or standard deviations
F* = s12/s22 where dfn = n1 – 1 and dfd = n2 – 1
For two-tailed tests, using the larger sample variance in the numerator will eliminate the necessity of calculating the left-tailed critical value.

For one-tailed left-tailed tests, using the larger sample variance in the numerator will turn the test into a one-tailed right-tailed test.

For left-tailed critical values, F(dfn, dfd, 1-α/2) = 1 / F(dfd, dfn, α/2)

(1 – α) Confidence Interval for σ12/σ22 is

s12/s22 to (s12/s22) F(df2, df1, α/2)
F(df1, df2, α/2)

1.5
Does the confidence interval you calculated give you EXACTLY the same information as the hypothesis test above?
A. No. The information is not the same and the decisions are not the same.
B. While the information is the same, the decisions are not the same.
C. Yes. The information is the same and the decisions are the same.
D. While the decisions are the same, the information is not exactly the same.

And what was your conclusion? - Why?

Any help on this would be greatly appreciated. I have worked into the problem this far and I am lost on 1.4. If you could check over the other components of the problem to see if I did them right, that would be great as well. I’m really struggling in Statistics lately. Thanks!
.
 
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