Math 135 Review Test 3

Math 135 Review Test 3

1. Suppose we know that the weight of male runners in the U.S. conforms to a normal distribution with a mean = 150 pounds and a standard deviation of 15 pounds.

1. If a single runner is selected at random, what is the probability that the runner will have a weight that is greater than 165 pounds?

1. Suppose a random sample of 40 runners is selected, what are the mean and standard deviation of the sampling distribution for x-bar?

1. Given that an SRS of 40 runners is selected, what is the probability that the average weight of the sample is greater than 165 pounds?

1. Doctors would like to be able to decrease the pulse rate of a patient during heart surgery. A medical study investigates the effect of a drug called a beta-blocker on the pulse rate of the patient during surgery. Researchers suspect that the new drug will reduce the mean pulse rate of the patient. Suppose that the true average pulse rate for a patient going through heart surgery is known to be 80 beats per minute. In the study, the pulse rates for the 16 people in the treatment group (those patients given the beta-blocker) are listed below. Assume the data were collected via an SRS and that the population is Normal.

7665826268877468816578

7480686570

a)Can we conclude that the beta blocker was successful? Set up the null and alternative hypotheses and construct an appropriate 95% confidence interval for the true mean pulse rate of the treatment group (those patients given the beta blocker) to defend your answer.

b)Conduct an appropriate significance test that is analogous to the 95% C.I. that you constructed in part a). Do you draw the same conclusion?

3. A study is conducted to investigate the room and board costs for American colleges. An SRS of 32 colleges is selected. The room and board costs (in $) of the sample of 32 are shown below.

a)Conduct a test to determine whether the true average room and board costs differs from $6000. Include the p-value in your answer. State your null and alternative hypotheses. Use a significance level of 0.02 to determine whether to reject Ho.

b)Construct a 98% confidence interval for the room and board costs. Would you draw the same conclusion as in part a? Explain.

c)How large would the sample size need to be if I wanted my 98% confidence interval to be no wider than  $500?

4. A study was conducted to determine whether skipping breakfast contributes to midmorning fatigue. The data from the survey is displayed in the following table:

Fatigue?

Ate Breakfast?

Yes

Assume the data were collected as an SRS.

a)Which variable is the explanatory variable? Which is the response variable?

b)Construct appropriate graphs to display the data in the table. (Construct the conditional distribution of the response variable given the explanatory variable).

c)Describe your graphs. Does there appear to be a relationship?

d)Conduct the appropriate significance test using α = 0.01. Do you have enough evidence to conclude that there is a relationship?

5. Dental researchers have developed a new material for preventing cavities – a plastic sealant that is applied to the chewing surfaces of teeth. To determine whether the sealant is effective, it was applied to half of the teeth of each of 12 school-age children (selected at random). After five years, the numbers of cavities in the sealant-coated teeth and untreated teeth were counted. The results are given in the table below. Is there sufficient evidence to indicate that sealant-coated teeth are less prone to cavities than untreated teeth? Conduct the appropriate significance test to defend your answer. Use a significance level of 0.01.

1. Tensile strengths were carried out on two different grades of wire rod (“Fluidized Bed Patenting of Wire Rods” Wire J. June 1977) resulting in the following data:

Assume that each sample was selected as an SRS from each “grade” population.

a) Does the data provide compelling evidence for concluding that the true average strength for the 1078 grade is different from the true average strength for the 1064 grade? Conduct an appropriate significance test using a significance level = .05.

b)Construct an appropriate confidence interval (that is analogous to the significance test conducted in part a)). Do you draw the same conclusion? Explain.

Answer Key:

1. a) .1587b) ; c)
2. a) CI: [68.96, 76.84]; Yes, evidence suggests the beta blocker was successful since the entire 95% CI is less than 80

b).0005< p-value<.001; Yes, the conclusion is again to reject

1. a) p-value = 0.3954; can’t reject

b) [5729.97, 6582.5]; can’t reject since 600 is in the 98% CI

c) minimum sample size of 24

1. a) Explanatory: Ate Breakfast Response: Fatigue

b)2 graphs; one for “Ate Breakfast”…%’s for Fatigue Y: 29% Fatigue N: 71%

one for “No Breakfast”…%’s for Fatigue Y: 41%; Fatigue N: 59%

c)Relationship seems to exist; Those who ate no breakfast were in general more likely to be fatigued

d) ; .005< p-value < .01; Since p-value is less than the given , we reject which means the evidence suggests a relationship exists

1. ; .025 < p-value < .05 which is not less than .01; not enough evidence to reject ; at there is not enough evidence to indicate the sealant coated teeth are less prone to cavities
2. a) ; p-value is essentially 0; Reject b) [15.69, 16.41]; yes, 0 is outside the 95% CI