Question

5. An application of the distribution of sample means

People suffering from hypertension, heart disease, or kidney problems may need to limit their intakes of sodium. The public health departments in some U.S. states and Canadian provinces require community water systems to notify their customers if the sodium concentration in the drinking water exceeds a designated limit. In Connecticut, for example, the notification level is 28 mg/L (milligrams per liter).

Suppose that over the course of a particular year the mean concentration of sodium in the drinking water of a water system in Connecticut is 26.4 mg / L, and the standard deviation is 6 mg / L.

Imagine that the water department selects a simple random sample of 32 water specimens over the course of this year. Each specimen is sent to a lab for testing, and at the end of the year the water department computes the mean concentration across the 32 specimens. If the mean exceeds 28 mg / L, the water department notifies the public and recommends that people who are on sodium-restricted diets inform their physicians of the sodium content in their drinking water.

Use the Distributions tool to answer the following question. (Hint: Start by setting the mean and standard deviation parameters on the tool to the expected mean and standard error for the distribution of sample mean concentrations.)

Even though the actual concentration of sodium in the drinking water is within the limit, there is a _______ probability that the water department will erroneously advise its customers of an above-limit concentration of sodium.

Suppose that the water department wants to reduce its risk of erroneously notifying its customers that the sodium concentration is above the limit. Assuming the water department can't change the mean or the standard deviation of the sodium concentration in the drinking water, is there anything the department can do to reduce the risk of notifying its customers that the sodium concentration is above the limit when it actually is not?

• It can collect more specimens over the course of the year.

• No, there is nothing it can do.

• It can collect fewer specimens over the course of the year.

Answer 1

a)
P(Xbar > 28)
Xbar = N( 26.4 , 6^2 / 32)

Z = (Xbar - 26.4)/ 1.06066

P(Xbar > 28)
= P(Z >   (28 - 26.4)/ 1.06066)
= P(Z >    1.5084947 )
= 0.0657

b)
A) it can collect more specimen