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Hypothesis testing
Course:- Applied Statistics
Reference No.:- EM13484

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Assignment Help >> Applied Statistics

Con dence interval for μ when σ is known

Suppose n = 9 people are selected at random from a large population. Assume the heights of the people in this population are normal, with mean μ = 68:71 inches and σ = 3 inches. Simulate the results of this selection 20 times and in each case nd a 90% con dence interval for μ. The following commands may be used:

MTB > random 9 c1-c20;

SUBC> normal 68.71 3.

MTB > zinterval 0.90 3 c1-c20

a. How many of your intervals contain μ?

b. What is the probability that 100 (not 20) such intervals would contain ?

c. Do all the intervals have the same width?

d. Suppose you constructed 80% intervals instead of 90%. W

e. How many of your intervals contained the value 71?

f. Suppose you took samples of size n = 4 instead of n = 9. Would you expect more or fewer intervals to contain 71? What about 68.71? What about the width of the intervals for n = 4: Would they be narrower or wider than for n = 9?

2. Con dence interval for μ when σ is NOT known

Command to get the 20 90% intervals:

MTB > random 9 c1-c20;

SUBC > normal 68.71 3.

MTB > tinterval 0.90 c1-c20

a. How many of your intervals contain μ?

b. Would you expect all 20 of the intervals to contain μ?

c. Do all the intervals have the same width?

d. Suppose you took 95% intervals instead of 90%. Would they be narrower or wider?|

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e. How many of your intervals contain the value

f. Suppose you took samples of size n = 64 instead of n = 9. Would you expect more or fewer intervals to contain 71?

What about the width of the intervals for n = 64: Would they be narrower or wider than for n = 9?

3. Hypothesis testing for μ when  is known

Imagine choosing n = 16 women at random from a large population and measuring their heights. Assume that the heights of the women in this population are normal with μ = 63:8 inches and μ = 3 inches. Suppose you then test the null hypothesis H0 : μ = 63:8 versus the alternative that Ha : μ = 63:8, using μ = 0:10. Assume  is known. Simulate the results of doing this test 30 times as follows:
MTB > random 16 c1-c30; SUBC > normal 63.8 3. MTB > ztest 63.8 3 c1-c30

a. In how many tests did you reject H0. That is, how many times did you make an \incorrect decision"?

b. Are the p-values all the same for the 30 tests?

c. Suppose you used = 0:001 instead of = 0:10. Does this change any of your decisions to reject or not?

In general, should the number of rejections increase or decrease if = 0:001 is used instead of = 0:10?

d. Now assume that the population really has a mean of  = 63, instead of 63.8, and carry out the above 30 simulations, (thus, use the above minitab commands with 'normal 63.8 3' changed to 'normal 63 3'. Once again, using = 0:10 and assuming  known, in how many
tests did you reject H0?

A rejection of H0 in part (a) is a \correct decision". True or False?

A rejection of H0 in part (d) is a \correct decision". True or False?

4. Hypothesis testing for μ when  is NOT known Repeat Question 3, using ttest instead of ztest, and answer parts (a), (b), and (c) again.

a. In how many tests did you reject H0. That is, how many times did you make an \incorrect decision"?

b. Are the p-values all the same for the 30 tests?

c. Suppose you used = 0:00008 instead of = 0:10. Does this change any of your decisions to reject or not?

In general, should the number of rejections increase or decrease if = 0:00008 is used instead of = 0:10?

5. A fast food franchiser is considering building a restaurant at a certain location. According to a nancial analysis, a site is acceptable only if the number of pedestrians passing the location averages more than 100 per hour. A random sample of 50 hours produced  x = 110 and s = 12 pedestrians per hour.

(a) Do these data provide sucient evidence to establish that the site is acceptable? Use = 0:05.

(b) What are the consequences of Type I and Type II errors? Which error is more expensive to make?

(c) Considering your answer in part (b), should you select to be large or small? Explain.

(d) What assumptions about the number of pedestrians passing the location in an hour are necessary for your hypothesis test to be valid?

6. An experiment was conducted to test the e ect of a new drug on a viral infection. The infection was induced in 100 mice, and the mice were randomly split into two groups of 50.

The rst group, the control group, received no treatment for the infection. The second group received the drug. After a 30-day period, the proportions of survivors, in the two groups were found to be 0.36 and 0.60, respectively.

(a) [6] Is there sucient evidence to indicate that the drug is e ective in treating the viral infection? Test at 5% signi cance level. (Make sure to state your null and alternative hypotheses.)

(b) [6] Use a 95% con dence interval to estimate the actual di erence in the cure rates, i.e. p1 p2, for the treatment versus the control groups. Based on this con dence interval can you conclude that the drug is e ective? Why?

7. In an investigation of pregnancy-induced hypertension, one group of women with this disor- der was treated with low-dose aspirin, and a second group was given a placebo. A sample consisting of 23 women who received aspirin has mean arterial blood pressure 111 mm Hg and standard deviation 8 mm Hg; a sample of 24 women who were given the placebo has mean blood pressure 109 mm Hg and standard deviation 8 mm Hg.

(a) At the 0.01 level of signi cance, test the null hypothesis that the two populations of women have the same mean arterial blood pressure. Justify any procedure you use.

(b) Construct a 99% con dence interval for the true di erence in population means. Does this interval contain the value 0? [3] Based on this con dence interval, what is you conclusion regarding the e ect of the two treatments on the blood pressure of pregnant
women?

8. A company is interested in o ering its employees one of two employee bene t packages.

A random sample of the company's employees is collected, and each person in the sample is asked to rate each of the two packages on an overall preference scale of 0 to 100. Results were

Do you believe that the employees of this company prefer, on the average, one package over the other? Explain.

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