Question

For small sample sizes, an alternate method of constructing a confidence interval is available. To implement this method, first add an additional 4 observations in the sample, 2 of which are successes and 2 of which are failures. Second, construct a new sample proportion Ônew for the new sample. Finally, construct a confidence interval using Pnew with the formula

Pnew ± 1.96 √Pnew (1 – Pnew)/n +4)

a. For the cell phone batteries, consider a sample of 5 in which 1 battery fails to last through 500 charges. Using the alternate method described, compute the value of Pnew

Answer 1

Using the plus-four method, the new sample proportion Pnew for the cell phone batteries scenario is 1/3, calculated from 3 successes and 9 total samples after adjusting the original sample size.

Step-by-step explanation:

When constructing a confidence interval for small sample sizes using the plus-four method, the sample size is effectively increased by 4, with 2 assumed successes and 2 assumed failures. This is helpful in cases where the regular confidence interval formula may not be adequate due to the small size of the sample. Given that in the original sample of 5 batteries, 1 battery failed, which is equivalent to 1 success in terms of failure. Using the plus-four adjustment, we calculate the new proportion, Pnew, for the adjusted sample with 6 successes (1 original success + 2 additional successes) and 9 total samples (5 original samples + 4 additional samples).

Therefore, the new sample proportion, Pnew, is calculated as:

Pnew = (x + 2) / (n + 4)
= (1 + 2) / (5 + 4)
= 3 / 9
= 1/3

This new proportion is used to construct a confidence interval with a more reliable estimate for small samples.