Suppose you are studying the speed of cars passing a certain point on Highway 10

Question

Suppose you are studying the speed of cars passing a certain point on Highway 101 and denote this random variable by X. Suppose that X N(,2) and suppose that both and 2 are unknown. Based on a sample of 28 observations resulting in X = 62.3 and S = 13.7 you would now like to do inference involving for = 0.1.

a) Find a lower, an upper and a 2-sided confidence interval for .

b) Based on the confidence intervals you just found, can you be 90% sure that is less than 66?

c) Based on the confidence intervals you found above, can you be 97% sure that is at least 57.7?

d) Perform the hypothesis test H0 : = 65 Ha : < 65 by computing the test statistic. What do you conclude?

e) Compute the P-value associated with this test and explain for which values of the null would be rejected and for which values it would not.

f) Conduct the hypothesis test by using one of the confidence intervals you have found earlier. Which of the confidence intervals are relevant for this test and what is the general rule as to which confidence interval is to be matched with what hypothesis test?

g) For which range of values of 0 would you reject the null hypothesis?

h) Compute the power of test for a = 64 and a = 60. Explain when the power of the test is larger and why that is the case.

Explanation / Answer

Given

mean= 62.3

SD= 13.7

n=28

Assuming Normal distribution.

a) CI = mean +-z(.95)*SD/sqrt(n)

= 62.3+-1.65*13.7/sqrt(28)

b) H0: mean>=66

Ha: mean<66

z= -(66-62.3)/(13.7/SQRT(28)) = -1.43

p-value= 0.07<0.1; so reject H0 and conclude that mean<66

c) Yes.since 97% CI IS:

& 57.7 IS WITHIN LIMITS

d) H0 : = 65 Ha : < 65

test stat z= -(65-62.3)/(13.7/SQRT(28))=-1.04

p-value=.148>.1; cant reject H0; mean is not less than 65

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