1.The following observations are from two independent random samples, drawn from normally distributed populations.

Sample 1	[54.45, 61.46, 67.88, 54.56, 60.01, 77.84]
Sample 2	[61.56, 66.59, 65.06, 60.55, 63.16]

Test the null hypothesis H0:21=22 against the alternative hypothesis HA:2122 .

a) Using the larger sample variance in the numerator, calculate the F test statistic.

Round your response to at least 3 decimal places.

b) The p-value falls within which one of the following ranges:

A. p-value > 0.50

B. 0.10 < p-value < 0.50

C. p-value < 0.10

c) What conclusion can be made at the 10% level of significance?

A. There is insufficicent evidence to reject the null hypothesis at the 10% level of significance, and therefore no significant evidence that the population variances are not equal to each other.

B. There is sufficient evidence to reject the null hypothesis at the 10% level of significance, and therefore evidence that the population variances are not equal to each other.

2. Identify each of the following statements as either true of false.

a)The sum of the hypothesized probabilities must be equal to 1.

b)In a goodness-of-fit test, the null hypothesis is that all of the hypothesized proportions are the same, against the alternate hypothesis that they are all different.

c) For goodness-of-fit tests, if the chi-square test statistic is greater than 1 then there is significant evidence against the null hypothesis.

d) If the null hypothesis is true, the chi-square test statistic will follow a chi-square distribution with k degrees of freedom, where k is the number of categories.

e)In a one-way table, each observed value can fall into only one of the given classifications.

3. Suppose there is a random sample of400 observations, divided into four groups. The table below summarizes the count of observations that were seen in each group.

Group 1	Group 2	Group 3	Group 4
40	40	280	40

We are interested in testing the null hypothesis H0:p1=p2=p3=p4=0.25 , against the alternative hypothesis HA:Atleastoneproportionisincorrect .

a) What is the expected count for each of the groups?

Expected:

b) What is the value of the test statistic?

Round your response to at least 2 decimal places.

c) What are the appropriate degrees of freedom?

d) What conclusion can be made at the 5% level of significance?

A. There is very strong evidence against the null hypothesis, and therefore it is rejected in favour of the alternative hypothesis thatat least one proportion is not equal to 0.25.

B. There is no significant evidence against the null hypothesis, and therefore there is no significantevidence that any of the proportions is not equal to 0.25.

4. Identify each of the following statements as either true of false.

a) In a one-way table, each observed value can fall into only one of the given classifications.

b) The sum of the hypothesized probabilities must be equal to 1.

c) If the null hypothesis is false, then large differences between the observed and expected values may exist.

d) The sum of the observed values in each of the cells can be greater than n, the total number of observations.

e) If the null hypothesis is true, the chi-square test statistic will follow a chi-square distribution with k degrees of freedom, where k is the number of categories.

5. Suppose there is a random sample of400 observations, divided into four groups. The table below summarizes the count of observations that were seen in each group.

Group 1	Group 2	Group 3	Group 4
160	40	40	160

We are interested in testing the null hypothesis H0:p1=p2=p3=p4=0.25 , against the alternative hypothesis HA:Atleastoneproportionisincorrect .

a) What is the expected count for each of the groups?

Expected:

b) What is the value of the test statistic?

Round your response to at least 2 decimal places.

c) What are the appropriate degrees of freedom?

d) What conclusion can be made at the 5% level of significance.

A. There is very strong evidence against the null hypothesis, and therefore it is rejected in favour of the alternative hypothesis thatat least one proportion is not equal to 0.25.

B. There is no significant evidence against the null hypothesis, and therefore there is no significantevidence that any of the proportions is not equal to 0.25.

6. Suppose there is a random sample of100 observations, divided intothree groups. The table below summarizes the count of observations that were seen in each group.

Group 1	Group 2	Group 3
54	17	29

We are interested in testing the null hypothesis H0:p1=0.5,p2=0.2,p3=0.3 , against the alternative hypothesis HA:Atleastoneproportionisincorrect .

a) What is the value of the test statistic?

Round your response to at least 2 decimal places.

b) What conclusion can be made at the 5% level of significance?

A. There is very strong evidence against the null hypothesis, and therefore it is rejected in favour of the alternative hypothesis thatat least one proportion is not correct.

B. There is no significant evidence against the null hypothesis, and therefore there is no significantevidence that any of the proportions is not correct.

7.Suppose there is a random sample of1,189 observations, divided intofour groups. The table below summarizes the count of observations that were seen in each group.

Group 1	Group 2	Group 3	Group 4
571	190	143	285

We are interested in testing the null hypothesis H0:p1=0.5,p2=0.2,p3=0.1,p4=0.2 .

a) What is the appropriate alternative hypothesis?

A. HA: All of the proportions are incorrect.

B. HA: All of the proportions are equal to each other.

C. HA: At least one of the proportions is incorrect.

b) What is the value of the test statistic?

Round your response to at least3 decimal places.

c) The p-value falls within which one of the following ranges

A. p-value > 0.10

B. 0.05 < p-value < 0.10

C. 0.025 < p-value < 0.05

D. 0.01 < p-value < 0.025

E. p-value < 0.01

d) What conclusion can be made at the 5% level of significance

A. There is no significant evidence against the null hypothesis, and therefore the null hypothesis is not rejected.

B. There is very strong evidence against the null hypothesis, and therefore it is rejected in favour of the alternative hypothesis.