7. An application of the distribution of sample means Aa Aa People suffering fro

Question

7. An application of the distribution of sample means Aa Aa 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 Ontario, for example, the notification level is 20 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 Ontario is 18 mg/L, and the standard deviation is 6 mg/L Imagine that the water department selects a simple random sample of 30 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 30 specimens. If the mean exceeds 20 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.) Normal Distribution Mean = 19.5 Standard Deviation = 0.85 10 12 18 20 24 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 is willing to accept (at most) a 1% risk of erroneously notifying its customers that the sodium concentration is above the limit. A primary cause of sodium in the water supply is the salt that is applied to roadways during the winter to melt snow and ice. If the water department can't control the use of road salt and 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 an erroneous notification to 1%? It can increase its sample size to n = 48 It can increase its sample size to n = 34 It can increase its sample size to n 49 No, there is nothing it can do

Explanation / Answer

First thing is to calculate probability that drinking water is within the limit, even than it will be errounsly advised to customers by water department.

Mean Concentration = 18mg/L

Standard deviaion = 6 mg /ml

Standard error of the sample mean = / sqrt(n) = 6/ sqrt(30) = 1.095

Probability = Pr(Sample Mean Concentration > 20 ml) = 1 - Pr(sample mean concenration < 20ml ; 18; 1.095)

Z = (20 -18)/ 1.095 = 1.825

so Probability = Pr (Z > 1.825) = 1 - 0.966 = 0.034 = 3.4%

Now that probaility is only 1%

so Z - value = 2.33

so Pr( sample mean concentration > 20 mg/L) = 0.99

so Z = 2.33 = (20 - 18) / (/ sqrt(n) ) = 2/ (6/ sqrt(n))

6/ sqrt(n) = 2/2.33

sqrt(n) = 6 * 2.33/2 = 7

n = 49

so sample size can be incresed to n = 49 option C is correct.

Related Questions

Navigate

• Browse (All)
• Browse 7
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.