Question 12.4 Please write or type answer very clearly and as detailed as possib

Question

Question 12.4 Please write or type answer very clearly and as detailed as possible Thanks! a sample mean 81 320 though we don't know the population stalualu ge, so it is reasonable to use the sample standard deviation as an a. / At the 5% level of significance, should we concludethat 35-year-olds North Dakota have lower incomes on average than the national avera State the null and alternative hypothesis, find the rejection regio state your conclusion in common English. North Dakota is b. Suppose that the true mean income of 35-year-olds in actually $23,600. For the decision rule found in part a, find the prob- bility of committing a Type II error. Suppose that the mean income of 35-year-olds in the United States is $24,000. A random sample of 100 35-year-olds in California results in a sample mean income of $24,600 and a sample standard deviation of S4000. Although we don't know the population standard deviation ?, the sample is large, so it is nable to use the sample standard deviation as an estimate of it. 12.4 a. At the 5% level of significance, should we conclude that 35-year-olds in California have a State the null and altèrnative hypotheses, find the rejection region, and state your conclusion in common English. Suppose that the true mean income of 35-year-olds in California is actu- ally $24,500. For the decision rule found in part a, find the probability ? of committing a Type II error. verage income than the national average? b. 12.5 In the preceding five years, entering students at a certain university had an . average SAT verbal score of 612 points. A simple random sample of 100 students taken from this year's class showed the average SAT verbal score for students to be 630 with a standard deviation of 80 points. Does this show an

Explanation / Answer

Solution:

a. Null Hypothesis (Ho): The mean income of 35 -year-olds in California is not higher than the national average.

Alternative Hypothesis (Ha): The mean income of 35-year-olds in California is higher than the national average.

Test Statistics

Z = (X-bar - µ)/ (?/sqrt(n))

Z = (24600-24000)/ (4000/sqrt(100))

Z = 1.5

Rejection region - If Z > Z (0.05) = 1.645, we reject Ho and accept otherwise.

Since test statistics is less than the critical value, we fail to reject Ho.

Therefore the mean income of 35-year-olds in California is not higher than the national average.

b. The respective Z-score with p > 0.05 is 1.645

1.645 = (X - 24000)/(4000/sqrt(100))

1.645 = (X - 24000)/400

1.645*400 + 24000 = X

X = 24658

P (reject Ho|µ = 24500, ? = 4000) = P (X ? 24658|µ = 24500, ? = 4000)

The respective Z-score with X = 24658 is

Z = (24658 - 24000)/(4000/sqrt(100))

Z = 0.40

Using Z-tables,

1 - P (Z ? 0.40) = 1 - 0.65542 = 0.34458

? = 0.65542

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