Question

Hydrologists collected water samples from a river for several years. They found 62 out of 648 samples exceeded the desired pH levels, with a 2% samples exceeded the desired pH levels, with a 90% confidence level. Which of the following is the most reasonable conclusion?
A. It is likely that the river water exceeds the desired pH level between 8% and 12% of the time studied.

B. It is likely that the river water exceeds the desired pH level between 10% and 12% of the time studied.

C. It is likely that the river water exceeds the desired pH level between 13% and 15% of the time studied.

D. It is likely that the river water exceeds the desired pH level between 11% and 15% of the time studied.

Answer 1

Answer:

Correct option is (A).

Step-by-step explanation:

Let p = proportion of water samples that exceeded the desired pH level.

A sample of size n = 648 is selected. Of these samples X = 62 exceeded the desired pH levels.

The confidence interval for the population proportion is given by:

$CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}\Rightarrow CI=\hat p\pm MOE$

The MOE or margin of  error is estimated difference between the true population parameter value and the sample statistic value.

The information provided is:

MOE = 0.02

$\hat p=\frac{X}{n}=\frac{6}{648}=0.096$

Compute the 90% confidence interval for the proportion of water samples that exceeds the desired pH level as follows:

$CI=\hat p\pm MOE\\=0.096\pm 0.02\\=(0.076, 0.116)\\\approx(8\%, 12\%)$

Thus, the 90% confidence interval for the proportion of water samples that exceeds the desired pH level is (8%, 12%).

This confidence interval implies that there is a 90% confidence that the river water exceeds the desired pH level between 8% and 12% of the time studied.

The correct option is (A).