The mean hourly wage for production employees in goods-producing industries is $

Question

The mean hourly wage for production employees in goods-producing industries is $20.78 (BLS June 2017). Suppose that we take a sample of 30 workers from the manufacturing industry to see if the mean hourly wage differs from the reported mean of $20.78 for goods producing industries. We find that the sample mean of the hourly wages for those 30 workers is $21 A6. The population of wages is normal, and has a known standard deviation of sigma = $2.40 per hour. Using = 05, test whether the manufacturing workers have wages that are greater than those in the goods producing industries. That is, test: H_0: mu lessthanorequalto $20.78 versus H_A: mu > $20.78 The Internal Revenue Service (IRS) gives a help-line for taxpayers to cell and get answer to. According to the IRS, callers using the IRS system will not wait on hold for more than 10 minutes before being able to talk to an IRS employee. A taxpayer advocate takes a sample of 50 callers and finds a mean warring time of 12 minutes before being able to talk to an IRS employee. Based on past results, the standard deviation of waiting time is known to be 8 minutes. Test whether the mean waiting time is only 10 minutes as claimed by the IRS. That is, test: H_0: mu lessthanorequalto 10 versus H_A: mu > 10. Use alpha = 05

Explanation / Answer

Problem 1. a)
Given that, population mean = µ = 20.78,
population standard deviation = = 2.40
sample mean = Xbar = 21.46
sample size = n = 30,

We need to test the hypothesis;

H0: µ 20.78 vs Ha: µ > 20.78

Now we have known standard deviation and sample size n = 30 hence we can use Z-test here,

Hence the test statistic Z-stat = (Xbar - µ) / ( / sqrt(n) )

Hence by substituting the values in the formula and calculating we will get,

Z-stat = ( 21.46 - 20.78 ) / ( 2.4 / sqrt(30) ) = 1.55

Let the given level of significance = = 0.05, so for = 0.05, the critical value of z is Z-critical = 1.96

This critical value is obtained from the standard normal probability table.

Now we reject null hypothesis if, Z-stat > Z-critical,, but here for this problem we have Z-stat (1.55) < Z-critical (1.96) hence we do not reject the null hypothesis.

Hence we can say that, we do not have sufficient evidence to reject our claim that mean hourly wage µ 20.78 .

1.b)

Given that, population mean = µ = 10,
population standard deviation = = 8,
sample mean = Xbar = 12
sample size = n = 50,

We need to test the hypothesis;

H0: µ 10 vs Ha: µ > 10

Now we have known standard deviation and sample size n = 50, hence we can use Z-test here,

Hence the test statistic Z-stat = (Xbar - µ) / ( / sqrt(n) )

Hence by substituting the values in the formula and calculating we will get,

Z-stat = ( 12 - 10) / ( 8 / sqrt(50) ) = 1.77

Let the given level of significance = = 0.05, so for = 0.05, the critical value of z is Z-critical = 1.96

This critical value is obtained from the standard normal probability table.

Now we reject null hypothesis if, Z-stat > Z-critical,, but here for this problem we have Z-stat (1.77) < Z-critical (1.96) hence we do not reject the null hypothesis.

Hence we can say that, we do not have sufficient evidence to reject our claim that mean waiting time µ 10 .

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