STAT 200 Week 4 Homework Problems

PART: A

Problem 1. The commuter trains on the Red Line for the Regional Transit Authority (RTA) in Cleveland,

OH, have a waiting time during peak rush hour periods of twelve (12) minutes.

(a) What is the random variable?

(b) Find the height of this uniform distribution.

(c) Find the probability of waiting between four and five minutes.

(d) Find the probability of waiting between three and eight minutes.

(e) Find the probability of waiting five minutes exactly.

Problem 2. Find the z-score corresponding to the given area. Remember, z is distributed as the standard

normal distribution with mean of μ = 0 and standard deviation σ = 1.

(a) The area to the left of z is 25%.

(b) The area to the right of z is 65%.

(c) The area to the left of z is 10%.

(d) The area to the right of z is 5%.

(e) Find the value for z such that the area between –z and z is 95%. (Hint: Drawing a picture

might help.)

(f) Find the value for z such that the area between –z and z is 99%. (Hint: Drawing a picture

might help.)

Problem 3. According to a recent study the mean blood pressure for people in China is 128 mmHg with a

standard deviation of 23 mmHg. Assume that blood pressure is normally distributed.

(a) What is the random variable?

(b) Find the probability that a person in China has blood pressure of 140 mmHg or more.

(c) Find the probability that a person in China has blood pressure of 135 mmHg or less.

(d) Is it unusual for a person in China to have a blood pressure of 135 mmHg or less? Why or

why not?

(e) Find the probability that a person in China has blood pressure between 120 and 125 mmHg.

(f) What is the 90th percentile blood pressure for the people in China? (That is, what is the blood

pressure which 90 percent of the people have less than?

Continuing with this problem, suppose now a sample of 15 people are chosen from the above normal

distribution and we are interested in the mean blood pressure of such a sample.

(g) Describe the distribution of the sample means: What is its shape? What is its mean? What is

its standard deviation?

(h) Find the probability that the sample mean blood pressure of 15 people in China is 140 mmHg

or more.

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(i) Would it be unusual to find a sample mean blood pressure of 15 people in China of 140

mmHg or more? Why or why not?

Problem 4. A Maytag dishwasher has a mean life of 12 years with an estimated standard deviation of

1.25 years. Assume the life of a dishwasher is normally distributed.

(a) What is the random variable?

(b) Find the probability that a dishwasher will last more than 15 years.

(c) Find the probability that a dishwasher will last less than 6 years.

(d) Find the probability that a dishwasher will last between 8 and 10 years.

(e) If you found a dishwasher that lasted less than 6 years, would you think that you have a

problem with the manufacturing process? Why or why not?

(f) Maytag only wants to replace free of charge 5% of all dishwashers. How long should the

manufacturer make the warranty period?

Continuing with this problem, suppose now a sample of 10 dishwashers are chosen from the above

normal distribution.

(g) Describe the distribution of the sample means: What is its shape? What is its mean? What is

its standard deviation?

(h) Find the probability that the sample mean of the dishwashers is less than 6 years.

(i) If you found the sample mean life of the 10 dishwashers to be less than 6 years, would you

think that you have a problem with the manufacturing process? Why or why not?

Problem 5. The mean yearly rainfall in Sydney, Australia, is 136.9 mm and the standard deviation is

69.4mm. Assume rainfall is normally distributed.

(a) What is the random variable?

(b) Find the probability that the yearly rainfall is less than 100 mm.

(c) Find the probability that the yearly rainfall is more than 240 mm.

(d) Find the probability that the yearly rainfall is between 140 and 250 mm.

(e) If a year has a rainfall less than 100mm, does that mean it is an unusually dry year? Why or

why not?

(f) What rainfall amount are 90% of all yearly rainfalls more than?

Problem 6. As it turns out, the actual data for annual rainfalls for Sydney, Australia are given in the table

below. Based on these data, was our assumption of normality for the previous problem a correct one?

Why or why not?

146.8

90.9

84.1

Table: Annual Rainfall in Sydney, Australia

383

90.9

178.1

267.5

95.5

139.7

200.2

171.7

187.2

184.9

55.6

133.1

271.8

135.9

71.9

156.5

70.1

99.4

180

58

110.6

2

47.5

97.8

122.7

58.4

154.4

173.7

118.8

88

84.6

171.5

254.3

185.9

137.2

138.9

96.2

85

45.2

74.7

264.9

113.8

133.4

68.1

156.4

Problem 7. A random variable is normally distributed. It has a mean of 245 and a standard deviation of

21.

(a) Suppose we take a random sample of size 10.

(i) What is the shape of the distribution of the sample means? Why?

(ii) What is the mean of the sample means, and what is standard deviation of the sample

means?

(iii) Find the probability that a sample mean is more than 241.

(b) Suppose we take a random sample of size 35.

(i) What is the shape of the distribution of the sample means? Why?

(ii) What is the mean of the sample means, and what is standard deviation of the sample

means?

(iii) Find the probability that a sample mean is more than 241.

(c) Compare your answers in (a)(iii) and (b)(iii). Why is one smaller than the other?

Problem 8. The mean cholesterol levels of women age 45-59 in Kenya, Africa is 5.1 mmol/l and the

standard deviation is 1.0 mmol/l. Assume that cholesterol levels are normally distributed.

(a) What is the random variable?

(b) Find the probability that a woman age 45-59 in Kenya has a cholesterol level above 6.2

mmol/l (considered a high level).

(c) Suppose doctors decide to test the woman’s cholesterol level again and average the two

values. Find the probability that this woman’s mean cholesterol level for the two tests is above 6.2

mmol/l.

(d) Suppose doctors being very conservative decide to test the woman’s cholesterol level a third

time and average the three values. Find the probability that this woman’s mean cholesterol level for the

three tests is above 6.2 mmol/l.

(e) If the sample mean cholesterol level for this woman after three tests is above 6.2 mmol/l,

what could you conclude?

PART 2A

Problem 9.

According to the February 2008 Federal Trade Commission report on consumer fraud and identity theft,

23% of all complaints in 2007 were for identity theft. In that year, in the state of ALASKA alone, there

were 321 complaints of identity theft out of 1,432 consumer complaints.

(a) What is the random variable?

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(b) What is the population parameter?

(c) Set up and run a test of hypothesis to determine if the data from ALASKA provides enough

evidence to show that ALASKA had a lower proportion of identity theft than 23%. Use a level of

significance of 5%. Specifically, do the following:

(i) Write down the null and alternative hypotheses.

(ii) State the type I and type II errors in this case and state a consequences of each type

of error in the context of this problem.

(iii) Verify that the assumptions necessary to apply the test are satisfied.

(iv) What is the test statistic and what is the p-value?

(v) State your conclusion.

Problem 10.

In 2008, across the United States, 1 child in 88 is diagnosed with Autism Spectrum Disorder (ASD).

But in the state of Arizona alone, a sample of 32,601 children was taken and 507 were diagnosed with

ASD.

(a) Set up and run a test of hypothesis to determine if there sufficient data to show that the

incident of ASD is more in Arizona than nationally? Use a level of significance of 1%. Specifically, do the

following:

(i) Write down the null and alternative hypotheses.

(ii) Verify that the assumptions necessary to apply the test are satisfied.

(iii) What is the test statistic and what is the p-value?

(iv) State your conclusion

(b) Using the sample data from Arizona, construct a 99% confidence interval for the true

population proportion in Arizona of children diagnosed with ASD. Make sure to check the assumptions

necessary to construct the interval. Interpret the interval.

Problem 11.

The economic dynamism, which is the index of productive growth in dollars for countries that are

designated by the World Bank as middle-income are in the table below. Countries that are considered

high-income have a mean economic dynamism of 60.29.

Table: Economic Dynamism ($) of Middle Income Countries

25.8057 37.4511

51.915 43.6952 47.8506 43.7178 58.0767

41.1648 38.0793 37.7251 39.6553 42.0265 48.6159 43.8555

49.1361 61.9281 41.9543 44.9346 46.0521 48.3652 43.6252

4

50.9866

59.1724

39.6282

33.6074

21.6643

(a) Set up and run a test of hypotheses to determine if the data show that the mean economic

dynamism of middle-income countries is less than the mean for high-income countries? Use a level of

significance of 5%. Specifically, do the following:

(i) Write down the null and alternative hypotheses.

(ii) Verify that the assumptions necessary to apply the test are satisfied.

(iii) What is the test statistic and what is the p-value?

(iv) State your conclusion.

(b) Using the data in the table which represents the index of productive growth in dollars for

countries that are designated by the World Bank as middle-income, construct a 95% confidence interval

for the true mean economic dynamism of middle-income countries. Make sure to check the

assumptions necessary to construct the interval. Interpret the interval.

Problem 12.

Maintaining your balance may get harder as you grow older. A study was conducted to see how steady

the elderly is on their feet. They had the subjects stand on a force platform and have them react to a

noise. The force platform then measured how much they swayed forward and backward, and the data

is in the table below.

Table: Forward/backward Sway (in mm) of Elderly Subjects

19

30

20

19

29

25

21

24

50

Set up and run a test of hypothesis to determine if the elderly sway more than the mean forward sway

of younger people, which is 18.125 mm? Use a level of significance of 5%. Specifically, do the

following:

(i) Write down the null and alternative hypotheses.

(ii) Verify that the assumptions necessary to apply the test are satisfied.

(iii) What is the test statistic and what is the p-value?

(iv) State your conclusion.

Problem 13.

Suppose you compute a confidence interval with a sample size of 100. What will happen to the

confidence interval if the sample size decreases to 80?

Problem 14.

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In 2013, Gallup conducted a poll and found a 95% confidence interval of the proportion of Americans

who believe it is the government’s responsibility for health care. Give a statistical interpretation of this

interval.

6

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