A. A researcher reads that in the population of US students, the average income

Question

A. A researcher reads that in the population of US students, the average income of students is $1500. In a random sample of 300 students, he finds that the mean monthly income is $1,445 with a standard deviation of $530. Conduct a one-tailed t-test to see if the mean in his sample is significantly (p<.05) lower than the population mean. What is the value of "t" that you find?

B. In the previous question, there were 300 respondents. Look at the t-test table in your book. If the t-score that was calculated was between 1.7 and 1.9, would you be able to reject the null hypothesis (p<.05) that the means of the sample and the population are the same? [Note: Because we have a prediction (that the sample might be less than the population) we use a one-tailed test.]

c. In the previous example, could you reject the null hypothesis with 99% confidence?

Explanation / Answer

(A)

Data:    

n = 300   

= 1500   

s = 530   

x-bar = 1445   

Hypotheses:    

Ho: 1500   

Ha: < 1500   

Decision Rule:    

= 0.05   

Degrees of freedom = 300 - 1 = 299

Critical t- score = -1.649965768   

Reject Ho if t < -1.649965768   

Test Statistic:    

SE = s/n = 530/300 = 30.59956427

t = (x-bar - )/SE = (1445 - 1500)/30.5995642670502 = -1.797411215

(B) p- value = 0.036639636   

Decision (in terms of the hypotheses):    

Since -1.797411215 < -1.649965768 we reject Ho and accept Ha

Conclusion (in terms of the problem):

There is evidence that the mean income < $1500

(C) Since 0.0366 > 0.01, we could not have rejected the null hypothesis with 99% confidence.

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