Top-Ten Review for 3rd Quartile of Course

Top-Ten Review for 3rd Quartile of Course

Use the table and spinner for the relevant problems that follow.

degrees of freedom 2 3 4 5 6 7

t-critical values 4.303 3.182 2.776 2.571 2.447 2.365

degrees of freedom 8 9 10 11 12 13

t-critical values 2.306 2.262 2.228 2.201 2.179 2.16

degrees of freedom 14 15 16 17 18 19

t-critical values 2.145 2.131 2.12 2.11 2.101 2.093

degrees of freedom 20 21 22 23 24 25

t-critical values 2.086 2.08 2.074 2.069 2.064 2.06 Table 1: t-Critical Values for 95% Confidence Intervals

1) What is the probability of landing on 3 five or more times in six spins?

2) What is the expected value of the spinner?

3) Suppose a random variable X is normal with a mean of 50 and a standard deviation of 5. What is the probability 40 < X < 60?

4) Suppose a random variable X is normal with a mean of 50 and a standard deviation of 5. What is the probability 42 < X < 53?

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5) Newscasters from Channel 6 survey club members at Weber’s Gun Range as part of a report on the proportion of Bell County residents who favor a ban on the production of high capacity magazines for firearms.

Discuss the report with the critical eye of a statistician.

6) Scientists collect 9 samples of soil from Bell County and measure the radioactivity, recording the mean annual effective dose of radioactivity (mrem). The scientists find an average of 21 mrem with a standard deviation of 4 mrem. Estimate the mean annual effective dose of radioactivity from soil in Bell County with 95% confidence.

7) A paint manufacturer claims the average drying time for its latex paint is 2 hours (120 minutes). A wholesaler suspects the claim is incorrect, so they randomly select 18 cans of the disputed paint and measure the drying times. The experiment obtains a sample mean of ¯ x = 115.9 minutes and a sample standard deviation of s = 9 minutes. Perform a hypothesis test.

8) Idea Enterprises receives 75% of its supplies from Totem Corp and 25%

of its supplies from Rail-Wright. 1% of the parts from each Totem Corp and 2% of the parts from Rail-Wright are defective. If a part is defective, what is the probability that the part is a Totem Corp part?

9) The Motion Picture Association of America claims the distribution below describes the ages of moviegoers, and the manager at Good Times Theater has observed the numbers below in a sample of 1,000 visitors. Does the proposed distribution describe the ages of moviegoers at Good Times Theater?

X 2 to 17 18 to 24 25 to 39 40 to 49 50 or older

P (X) 27% 20% 20% 14% 20%

Observed Sample 240 214 183 156 207

Table 2: Ages of Moviegoers

10) The tread life of a tire is a normally distributed variable. Brand A cost

$100 per tire and has mean tread life of 40,000 miles with a standard

deviation of 750 miles. Brand Z cost $85 per tire and has mean tread life of 40,000 miles with a standard deviation of 8,000 miles. Which tire would you buy and why?

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Answers

1) The spinner could land on 3 five times or six times. p =

+

=

2) µ = 2.5

3) By the 2SD rule, p = 0.95.

4) p ≈ 0.6709

5) A cluster sample selects members of a population that appear in clusters. Bell County residents at a gun range represent a cluster sample because the gun range members all share a common interest. Any report based on a cluster sample will introduce sampling bias. Sampling bias is any bias produced by the sampling procedure that is not due to random selection. The gun range members all share an interest in guns and are more likely to oppose any restriction on guns or ammunition magazines.

Thus, the cluster sample is biased against the ban.

6) margin of error = 2.306 ·

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≈ 3.075 Estimate = (17.925, 24.075) 7) H

: µ = 120 H

: µ 6= 120 α = 0.05 df = 17 t ≈ −1.933

p ≈ 2 × 0.035 ≈ 0.07 Fail to reject the null at 5% significance.

8) p =

= 0.6.

9) H

: the Motion Picture Association of America distribution of ages describes the ages of moviegoers at Good Times Theater. H

: the Motion Picture Association of America distribution of ages is incorrect for at least one age category for moviegoers at Good Times Theater. α = 0.05

χ

= 7.779 p ≈ 0.1 Fail to reject the null at 5% significance.

10) Answers will vary but should discuss the 2SD Rule. I would spend the extra cash and buy Brand A. Then, I can be 95% confident that the tire will have a tread life between 38,500 and 41,500 miles (which means I can be 97.5% confident that the tread life will last more than 38,500 miles). On the other hand, if I were to buy the cheaper Brand X tires, I could be 95%

confident that the tires will have a life from 24,000 to 56,000 miles. A gambler might hope for the chance the tires last for the upper end of this range, but the variability of this tire makes me think the tire is unreliable.

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