Question

Researchers presented a random sample of undergraduates with two test sheets, each sheet including 20 photos of the faces of dog‑owner pairs taken at a dog‑lovers field festival. The sets of dog‑owner pairs on the two sheets were equivalent with respect to breed, diversity of appearance, and gender of owners. On one sheet, the dogs were matched with their owners, while on the second sheet, the dogs and owners were deliberately mismatched. Students were asked to "choose the set of dog‑owner pairs that resemble each other, Sheet 1 or Sheet 2 ," and were simply told the aim of the research was a "survey on dog‑owner relationships." Of the 61 student judges in this part of the study, 49 chose the sheet with dog‑owners correctly matched. The large‑sample confidence interval for the proportion of undergraduates who can choose the sheet with the dogs and owners correctly matched was computed, although the conditions for the large‑sample confidence interval were not quite met.

(a) Are the conditions for the use of the plus four confidence interval met? If so, use the plus four method to give a 99% confidence interval for the proportion of subjects who can choose the sheet with the dogs and owners correctly matched.

Provided we have an SRS, the conditions are met. The plus four 99% confidence interval is about 0.650 to 0.920 .

Provided we have an SRS, the conditions are met. The plus four 99% confidence interval is about 0.654 to 0.916 .

No, even assuming we have an SRS, the conditions are not met.

Provided we have an SRS, the conditions are met. The plus four 99% confidence interval is about 0.685 to 0.885 .

(b) If you express the interval as a percent, round to the nearest 10th of a percent and compare the interval in part

(a) with the large‑sample 99% confidence interval computed to be about 0.6722 to 0.9344 . (As always, the plus four method pulls results away from 0% or 100%, whichever is closer.) As a percent, the interval is from 65.4% to 91.6% . This interval is shifted slightly—less than the previously computed interval by about 1.8 percentage points. As a percent, the interval is from 65.4% to 91.6% . This interval is shifted slightly—greater than the previously computed interval at both endpoints, and also wider. As a percent, the interval is from 65.4% to 91.6% . This interval is shifted slightly—less than the previously computed interval at both endpoints, and also narrower. As a percent, the interval is from 65.4% to 91.6% . This interval is shifted slightly—greater than the previously computed interval by about 1.8 percentage points.

Answer 1

a) The conditions for the use of the plus four confidence interval met since the confidence level is at least 90% and the sample size is at least 10.

The confidence interval for proportion is

. Here

The 99% Plus 4 confidence interval is  

Provided we have an SRS, the conditions are met. The plus four 99% confidence interval is about 0.654 to 0.916 .

b) The interval is expressed as percent. .  The large‑sample 99% confidence interval computed to be about 0.6722 to 0.9344 .

This interval is shifted slightly—greater than the previously computed interval at both endpoints, and also wider. As a percent, the interval is from 65.4% to 91.6%