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

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7. An application of the distribution of sample means Aa Aa E 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.) Normal Distribution Mean = 25.5 Standard Deviation 0.85 20 24 26 28 30 32 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 196? O No, there is nothing it can do It can increase its sample size to n = 76 It can increase its sample size to n = 48 It can increase its sample size to n = 88

Explanation / Answer

here std error of mean =std deviaiton/(n)1/2 =6/(32)1/2 =1.06

probability of exceeding limit =P(X>28) =P(Z>( 28-26.4)/1.06)=P(Z>1.5085)=0.0657

2) for reducing rik to 1% ; z score =2.326

corresponding margin of error =E =28-26.4=1.6

hence std error =E/z =0.6877

hence sample size n =(std deviation/std error)2 =(6/0.6877)2 =~76

therefore option : it can increase its sample size to n=76 is correct

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