Question

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Q8: New York is known as "the city that never sleeps". Suppose the amount of sleep (in hours) is nearly normal. A random sample of 25 New Yorkers were asked how much sleep they get per night. Statistical summaries of these data are shown below. Do these data provide strong evidence that New Yorkers sleep less than 8 hours a night on average? Assume the significant level is α-0.05. in max 257.73 0.77 6.179.78 Step 1: Parameter of interest: average amount of sleep of New Yorkers (mean of one population). Step 2: What is the null hypothesis? Step 3: What is the alternative hypothesis? Step 4: What is the test statistic? Step 5: We will reject Ho if the p-value the significance level α. (Fill the blank using " ", "-", Step 6: Step 7: Compute the test statistic based on the current sample. Compute the p-value, and make a decision. For the above problem Q8, answer the following questions. (a) What is the type I error rate of the test if the significance level is α? (b) What is the power of the test if the sample is i.i.d. from N(7.5,1)? What is the power of the test if the sample is 1.1.d. from N(7,1)? Assume the significant level is α 0.05.

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Answer 1

a)

The type I error rate or significance level is the probability of rejecting the null hypothesis given that it is true. It is denoted by the Greek letter α (alpha) and is also called the alpha level.

hence type i error is α

b)

power of test is rejecting the null when it is not true

we reject the null when
Xbar < 7.73 + t * s /sqrt(n)
here
s = 0.77
n = 25
df = n-1 = 24
t =   =T.INV(0.05,24)
= -1.71088

Xbar < 7.73 - 1.71088 * 0.77/sqrt(25)
< 7.46652448

power when sample is from N(7.5,1)
P(Xbar < 7.46652448 | N(7.5,1))
= P(Z <(7.46652448 - 7.5)/(1/sqrt(25)))
= P(Z < -0.1673776 )
= 0.433536475

when sample is from N(7,1)

P(Xbar < 7.46652448 | N(7,1))
= P(Z <(7.46652448 - 7)/(1/sqrt(25)))
= P(Z < 2.3326224)
= 0.990166

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