Question

Acid rain, caused by the reaction of certain air pollutants with rainwater, is a growing problem in the United States. Pure rain falling through clean air registers a pH value of 5.7 (pH is a measure of acidity: 0 is acid; 14 is alkaline). Suppose water samples from 80 rainfalls are analyzed for pH, and x and s are equal to 3.1 and 0.4, respectively. Find a 99% confidence interval for the mean pH in rainfall. (Round your answers to three decimal places.)

Interpret the interval:

A. In repeated sampling, 1% of all intervals constructed in this manner will enclose the population mean.

B. There is a 1% chance that an individual sample mean will fall within the interval.

C. In repeated sampling, 99% of all intervals constructed in this manner will enclose the population mean.

D. There is a 99% chance that an individual sample mean will fall within the interval.

E. 99% of all values will fall within the interval.

What assumption must be made for the confidence interval to be valid?

A. The sampling distribution must be symmetrical.

B. The sample must be random.

C. There must be at least 100 samples.

D. The sample mean must be greater than 5.

E. The standard deviation must be less than 10.

Answer 1

Here, sample mean = 3.1 , s = 0.4 , n= 80

The t value at 99% confidence interval is,

alpha = 1 - 0.99 = 0.01
alpha/2 = 0.01/2 = 0.005
t(alpha/2,df) = t(0.005,79) = 2.640

Margin of error = E =z *(s/sqrt(n))
= 2.640 *(0.4/sqrt(80))
= 0.118

The 95% confidence interval is

mean -E < mu < mean +E

3.1 - 0.118 < mu < 3.1+ 0.118

2.982 < mu < 3.218

Interpretation:

D. There is a 99% chance that an individual sample mean will fall within the interval.

A. The sampling distribution must be symmetrical.

B. The sample must be random.