Question

A report says that 82% of British Columbians over the age of 25 are high school graduates. A survey of randomly selected residents of a certain city included 1290 who were over the age of 25, and 1135 of them were high school graduates. Does the city's survey result provide sufficient evidence to contradict the reported value, 82%?

Part i) What is the parameter of interest?

A. The proportion of 1290 British Columbians (aged above 25) who are high school graduates.
B. Whether a British Columbian is a high school graduate.
C. All British Columbians aged above 25.
D. The proportion of all British Columbians (aged above 25) who are high school graduates.

ii) Let pp be the population proportion of British Columbians aged above 25 who are high school graduates. What are the null and alternative hypotheses?

A. Null: p=0.82p=0.82. Alternative: p≠0.82p≠0.82 .
B. Null: p=0.82p=0.82. Alternative: p=0.88p=0.88.
C. Null: p=0.88p=0.88. Alternative: p>0.88p>0.88.
D. Null: p=0.82p=0.82. Alternative: p>0.82p>0.82.
E. Null: p=0.88p=0.88. Alternative: p≠0.82p≠0.82.
F. Null: p=0.88p=0.88. Alternative: p≠0.88p≠0.88.

Part iii) The PP-value is less than 0.0001. Using all the information available to you, which of the following is/are correct? (check all that apply)

A. The observed proportion of British Columbians who are high school graduates is unusually low if the reported value 82% is incorrect.
B. The observed proportion of British Columbians who are high school graduates is unusually high if the reported value 82% is correct.
C. Assuming the reported value 82% is incorrect, it is nearly impossible that in a random sample of 1290 British Columbians aged above 25, 1135 or more are high school graduates.
D. The observed proportion of British Columbians who are high school graduates is unusually high if the reported value 82% is incorrect.
E. The reported value 82% must be false.
F. Assuming the reported value 82% is correct, it is nearly impossible that in a random sample of 1290 British Columbians aged above 25, 1135 or more are high school graduates.
G. The observed proportion of British Columbians who are high school graduates is unusually low if the reported value 82% is correct.

Part iv) Based on the PP-value (less than 0.0001) obtained, at the 5% significance level, ...

A. we should not reject the null hypothesis.
B. we should reject the null hypothesis.

Part v) What is an appropriate conclusion for the hypothesis test at the 5% significance level?

A. There is sufficient evidence to contradict the reported value 82%.
B. There is insufficient evidence to contradict the reported value 82%.
C. There is a 5% probability that the reported value 82% is true.
D. Both A. and C.
E. Both B. and C.

Part vi) Which of the following scenarios describe the Type II error of the test?

A. The data suggest that reported value is incorrect when in fact the value is correct.
B. The data suggest that reported value is correct when in fact the value is correct.
C. The data suggest that reported value is incorrect when in fact the value is incorrect.
D. The data suggest that reported value is correct when in fact the value is incorrect.

Part vii) Based on the result of the hypothesis test, which of the following types of errors are we in a position of committing?

A. Type I error only.
B. Both Type I and Type II errors.
C. Type II error only.
D. Neither Type I nor Type II errors.

Answer 1

I Given : n= 12go; x = 1135 1135 0.8798 n 1290 part-lo the proportion of all british Columbians Caged above 25) who are high school graduates. part-2: Null: p=0.82. Alternative: 0.82 part 3: Test statistic > p-p PCI-D) 2 - DOT 0.8798-0.82 0.82 X 0.18 12. 90 0.0598 2 0.0107 2 - 5.59 : 0:0538 p-vwue = 0.00001 (from standard normal part 3: As p-value cooool part (€) and (b) are correct part 4: we should reject the null hypothesis. part 5 : to contradict There is sufficient evidence the reported value 82 4. option cd) Type I error only part 6: part 7: