Question 1.

We will measure the speeds (in km/h) of a random sample of 37 vehicles driving on a

highway and conduct a hypothesis test at the 1% level of significance to determine

whether the true mean speed of all vehicles on this highway exceeds 100 km/h. The

standard deviation of speeds of all vehicles on the highway is known to be 15.4 km/h.

*Give all numerical answers to four decimal places.

(a) What is the rejection region in terms of

x

?

(b) What is the power of the test if the true mean speed of all vehicles on this highway

is actually 110 km/h?

(c) What is the probability of making a Type II Error if the true mean speed of all

vehicles on this highway is actually 110 km/h?

(d) Suppose we had instead used a 5% level of significance to conduct the test. What

x

is the rejection region in terms of ?

(e) Using a 5% level of significance, what is the power of the test if the true mean

speed of all vehicles on this highway is actually 110 km/h?

(f) What happens to the power of a test if we increase the probability of a Type I

error? (increase or decrease)

Question 2

The labels on bags of a certain brand of tortilla chips claim the bags contain 14 ounces

of chips. Weights actually vary slightly from bag to bag. Weights are known to

follow a normal distribution with standard deviation 0.28 ounces. An employee of a

consumer advocate group would like to see if there is evidence that the true mean

weight of all bags is less than advertised. To do this, he selects a random sample of 9

bags of tortilla chips and measures the weight of each.

*Give all numerical answers to four decimal places.

(a) Using a 10% level of significance, what is the power of the test if the true mean is

actually 13.8 ounces?

(b) Using a 10% level of significance, what is the power of the test if the true mean is

actually 13.7 ounces?

(c) Comparing your answers for (a) and (b), what happens to the power of the test

when you increase the size of the difference you want to detect? (increase or decrease)

(d) Suppose we had used a sample size of 25. Using a 10% level of significance,

what is the power of the test if the true mean is acutally 13.8 ounces?

(e) Comparing your answers for (a) and (d), what happens to the power of the test if

we increase the sample size? (Does it increase or decrease?)

3. A company claims that the amount of nicotine in their brand of cigarettes is 8.5 mg.

Nicotine content for this brand of cigarettes is known to follow a normal distribution

with standard deviation 0.89 mg. We will select a random sample of 12 cigarettes and

conduct a hypothesis test at the 10% level of significance to determine whether the

true mean nicotine content of all cigarettes differs from 8.5 mg.

*Give all answers to four decimal places.

(a) What is the rejection rule in terms of

x

?

(b) Using a 10% level of significance, what is the power of the test if the true mean

nicotine content is actually 8.8?

*Note: There is a much easier way to do part (c) than doing a power calculation from

scratch.

(c) Using a 10% level of significance, what is the power of the test if the true mean

nicotine content is actually 8.2?

4. An art collector is trying to decide whether to pay $2,000 to buy insurance for a

$20,000 painting. She is essentially testing the hypotheses

H : The painting will not be stolen vs. Ha: The painting will be stolen

0

Explain what it would mean in this case to make a Type I error and a Type II error.

Explain the potential consequences of each wrong decision.

5. An automobile manufacturer claims that the average mileage of its vehicles is

greater than 32 miles per gallon (mpg). We take a random sample of 20 vehicles and

we measure the mileage of each of them. The cars in the sample have a mean mileage

of 33.3 mpg and a standard deviation of 1.9 mpg. Mileages are known to follow a

normal distribution with standard deviation 2.5 mpg. We will conduct a hypothesis

test at the 5% level of significance to investigate the manufacturer's claim.

(a) What is the appropriate test statistic for this test? (z or t?) Explain why.

(b) What is the value of the test statistic for the appropriate test of significance?

(c) What is the P-value of the appropriate test of significance?

(d) What is the conclusion for the test?

(e) Suppose we had instead used the critical value method to conduct the test in (a).

What would be the rejection rule and the conclusion?

6. Environmentalists measure the pH levels of 30 samples of acid rain in a certain

region. The data are as follows:

4.8 4.4 5.0 4.9 5.6 4.3 5.2 4.9 5.4 4.4 4.0 4.6 5.3 4.4 5.7

4.8 4.3 4.6 4.1 5.0 4.9 4.2 4.5 5.3 4.9 4.6 4.7 5.1 4.9 5.2

The population standard deviation of acid rain pH levels is known to be 0.54.

(a) We have no knowledge about whether pH levels follow a normal distribution.

Why is it nevertheless appropriate to use inference methods which rely on the

assumption of normality?

(b) Construct a 93% confidence interval for the true mean pH level of acid rain in the

region. Explain how you find the critical value z*.

(c) Provide an interpretation of the interval in (b).

(d) Conduct a hypothesis test at the 7% level of significance to determine whether the

true mean pH level of acid rain in a certain region differs from 5.0. Use the critical

value method. Show all of your steps, including the hypotheses, the decision rule, the

calculation of the test statistic, and a properly worded conclusion.

(e) Suppose you had used the P-value method to conduct the test in (d). What is the

P-value of the test? What would be your conclusion, and why?

(f) Interpret the meaning of the P-value you calculated in (e).

(g) Verify your results for the test using JMP. Type the data values in a column

titled pH level. Go to Analyze > Distribution. Under the red arrow, select Test

Mean. Enter 5 for the hypothesized mean and 0.54 as the true standard

deviation. You do not need to attach the output to your assignment. What is the exact

P-value of the test, as given by JMP?

(h) Could the interval in (b) have been used to conduct the test in (d)? If no, explain

why not. If yes, explain why, and explain what your conclusion would have been and

why

7. An experimenter is concerned about radiation levels in her research laboratory.

Radiation levels are known to follow a normal distribution. A random sample of 12

radiation measurements is taken at different locations in the laboratory, the values of

which are shown below:

42 41 39 43 38 42 41 40 39 41 39 44

7 0 7 3 9 6 2 8 4 5 9 6

The mean and standard deviation for these twelve measurements are 413 and 17.273,

respectively.

(a) Calculate a 90% confidence interval for the true mean radiation level in the

laboratory.(Round values to two decimal places.)

(b) A room is only considered safe to work in if the mean radiation level is 400 or

lower. The experimenter would like to conduct a hypothesis test at the 10% level of

significance to determine if the laboratory is unsafe. What are the appropriate

hypotheses for the appropriate test of significance?

(c) What is the value of the appropriate test statistic?

(d) What is the P-value of the test?

(e) Use JMP to verify your calculations. To conduct a t test in JMP, enter the data

values into a column titled Radiation Level. Select Analyze > Distribution,

click Radiation Level, then Y, Columns, then OK. Under the red arrow, select Test

Mean. Enter 400 as the hypothesized mean, and leave the standard deviation box

empty. (This tells JMP we don't know the population standard deviation, so it will

know to do a t test rather than a z test.) You do not need to attach the output to your

assignment. What is the exact P-value of the test (to four decimal places)?

(f) What is the conclusion of the test?

(g) Suppose you had instead used the critical value method to conduct the test.

What would be the decision rule and conclusion?

8. We take a random sample of 22 students at a large university. The mean and

standard deviation of the GPAs of these 22 students are calculated to be 3.01 and 0.44,

respectively. It is known that GPAs at the university follow a normal distribution.

(a) Conduct a hypothesis test, using the P-value method, at the 5% level of

significance to determine whether the mean GPA of students at this university differs

from 3.2. Show all of your steps and calculations.

(b) Interpret the P-value of the test in (a).

(c) Suppose you had instead conducted the test using the critical value method. What

would be the decision rule and the conclusion?