1.) Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 73 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean μ = 73 tons and standard deviation σ = 1.1 ton.

(a) What is the probability that one car chosen at random will have less than 72.5 tons of coal? (Round your answer to four decimal places.)

(b) What is the probability that 20 cars chosen at random will have a mean load weight x of less than 72.5 tons of coal? (Round your answer to four decimal places.)

(c) Suppose the weight of coal in one car was less than 72.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment?

YesNo

Suppose the weight of coal in 20 cars selected at random had an average x of less than 72.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?

Yes, the probability that this deviation is random is very small.Yes, the probability that this deviation is random is very large.    No, the probability that this deviation is random is very small.No, the probability that this deviation is random is very large.

 

2.) Let x be a random variable that represents white blood cell count per cubic milliliter of whole blood. Assume that x has a distribution that is approximately normal, with mean μ = 6350 and estimated standard deviation σ = 2600. A test result of x < 3500 is an indication of leukopenia. This indicates bone marrow depression that may be the result of a viral infection.

(a) What is the probability that, on a single test, x is less than 3500? (Round your answer to four decimal places.)

(b) Suppose a doctor uses the average x for two tests taken about a week apart. What can we say about the probability distribution of x?

The probability distribution of x is not normal.The probability distribution of x is approximately normal with μx = 6350 and σx = 1838.48.    The probability distribution of x is approximately normal with μx = 6350 and σx = 1300.00.The probability distribution of x is approximately normal with μx = 6350 and σx = 2600.

What is the probability of x < 3500? (Round your answer to four decimal places.)

(c) Repeat part (b) for n = 3 tests taken a week apart. (Round your answer to four decimal places.)

(d) Compare your answers to parts (a), (b), and (c). How did the probabilities change as n increased?

The probabilities stayed the same as n increased.The probabilities increased as n increased.    The probabilities decreased as n increased.

If a person had x < 3500 based on three tests, what conclusion would you draw as a doctor or a nurse?

It would be an extremely rare event for a person to have two or three tests below 3,500 purely by chance. The person probably does not have leukopenia.It would be an extremely rare event for a person to have two or three tests below 3,500 purely by chance. The person probably has leukopenia.    It would be a common event for a person to have two or three tests below 3,500 purely by chance. The person probably has leukopenia.It would be a common event for a person to have two or three tests below 3,500 purely by chance. The person probably does not have leukopenia.

3.) Let x be a random variable that represents the weights in kilograms (kg) of healthy adult female deer (does) in December in a national park. Then x has a distribution that is approximately normal with mean μ = 56.0 kg and standard deviation σ = 7.8 kg. Suppose a doe that weighs less than 47 kg is considered undernourished.

(a) What is the probability that a single doe captured (weighed and released) at random in December is undernourished? (Round your answer to four decimal places.)

(b) If the park has about 2850 does, what number do you expect to be undernourished in December? (Round your answer to the nearest whole number.)
does

(c) To estimate the health of the December doe population, park rangers use the rule that the average weight of n = 40 does should be more than 53 kg. If the average weight is less than 53 kg, it is thought that the entire population of does might be undernourished. What is the probability that the average weight

x

for a random sample of 40 does is less than 53 kg (assuming a healthy population)? (Round your answer to four decimal places.)

(d) Compute the probability that

x

< 57.6 kg for 40 does (assume a healthy population). (Round your answer to four decimal places.)

Suppose park rangers captured, weighed, and released 40 does in December, and the average weight was

x

= 57.6 kg. Do you think the doe population is undernourished or not? Explain.

Since the sample average is below the mean, it is quite likely that the doe population is undernourished.Since the sample average is above the mean, it is quite unlikely that the doe population is undernourished.    Since the sample average is below the mean, it is quite unlikely that the doe population is undernourished.Since the sample average is above the mean, it is quite likely that the doe population is undernourished.

4.) Allen’s hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 20 Allen’s hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen’s hummingbirds have a normal distribution, with σ = 0.28 gram.

(a) Find an 80% confidence interval for the average weights of Allen’s hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b) What conditions are necessary for your calculations? (Select all that apply.)

σ is knownuniform distribution of weightsnormal distribution of weightsn is largeσ is unknown

(c) Interpret your results in the context of this problem.

The probability to the true average weight of Allen’s hummingbirds is equal to the sample mean.The probability that this interval contains the true average weight of Allen’s hummingbirds is 0.80.    The probability that this interval contains the true average weight of Allen’s hummingbirds is 0.20.There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen’s hummingbirds in this region.There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen’s hummingbirds in this region.

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.07 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)
hummingbirds

5.) Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma.† Over a period of months, an adult male patient has taken nine blood tests for uric acid. The mean concentration was x = 5.35 mg/dl. The distribution of uric acid in healthy adult males can be assumed to be normal, with σ = 1.85 mg/dl.

(a) Find a 95% confidence interval for the population mean concentration of uric acid in this patient’s blood. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b) What conditions are necessary for your calculations? (Select all that apply.)

n is largenormal distribution of uric acidσ is knownuniform distribution of uric acidσ is unknown

(c) Interpret your results in the context of this problem.

There is a 95% chance that the confidence interval is one of the intervals containing the population average uric acid level for this patient.The probability that this interval contains the true average uric acid level for this patient is 0.95.    There is a 5% chance that the confidence interval is one of the intervals containing the population average uric acid level for this patient.There is not enough information to make an interpretation.The probability that this interval contains the true average uric acid level for this patient is 0.05.

(d) Find the sample size necessary for a 95% confidence level with maximal margin of error E = 1.06 for the mean concentration of uric acid in this patient’s blood. (Round your answer up to the nearest whole number.)
blood tests

6.) Thirty-one small communities in Connecticut (population near 10,000 each) gave an average of x = 138.5 reported cases of larceny per year. Assume that σ is known to be 44.7 cases per year.

(a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit

upper limit

margin of error

 
 

(b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit

upper limit

margin of error

(c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

lower limit

upper limit

margin of error

(d) Compare the margins of error for parts (a) through (c). As the confidence levels increase, do the margins of error increase?

As the confidence level increases, the margin of error decreases.As the confidence level increases, the margin of error increases.    As the confidence level increases, the margin of error remains the same.

(e) Compare the lengths of the confidence intervals for parts (a) through (c). As the confidence levels increase, do the confidence intervals increase in length?

As the confidence level increases, the confidence interval remains the same length.As the confidence level increases, the confidence interval increases in length.    As the confidence level increases, the confidence interval decreases in length.

7.) How much does a sleeping bag cost? Let’s say you want a sleeping bag that should keep you warm in temperatures from 20°F to 45°F. A random sample of prices ($) for sleeping bags in this temperature range is given below. Assume that the population of x values has an approximately normal distribution.

35
70
85
110
105
55
30
23
100
110

105
95
105
60
110
120
95
90
60
70

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean price x and sample standard deviation s. (Round your answers to two decimal places.)

(b) Using the given data as representative of the population of prices of all summer sleeping bags, find a 90% confidence interval for the mean price μ of all summer sleeping bags. (Round your answers to two decimal places.)

lower limit
$

upper limit
$

 
 

8.) Do you want to own your own candy store? Wow! With some interest in running your own business and a decent credit rating, you can probably get a bank loan on startup costs for franchises such as Candy Express, The Fudge Company, Karmel Corn, and Rocky Mountain Chocolate Factory. Startup costs (in thousands of dollars) for a random sample of candy stores are given below. Assume that the population of x values has an approximately normal distribution.

99
172
133
97
75
94
116
100
85

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean startup cost x and sample standard deviation s. (Round your answers to one decimal place.)

x =
 thousand dollars

s =
 thousand dollars

(b) Find a 90% confidence interval for the population average startup costs μ for candy store franchises. (Round your answers to one decimal place.)

lower limit
 thousand dollars

upper limit
 thousand dollars

9.) Over the past several months, an adult patient has been treated for tetany (severe muscle spasms). This condition is associated with an average total calcium level below 6 mg/dl. Recently, the patient’s total calcium tests gave the following readings (in mg/dl). Assume that the population of x values has an approximately normal distribution.

10.1
9.4
10.7
9.5
9.4
9.8
10.0
9.9
11.2
12.1

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean reading x and the sample standard deviation s. (Round your answers to two decimal places.)

x =
 mg/dl

s =
 mg/dl

(b) Find a 99.9% confidence interval for the population mean of total calcium in this patient’s blood. (Round your answer to two decimal places.)

lower limit
 mg/dl

upper limit
 mg/dl

(c) Based on your results in part (b), do you think this patient still has a calcium deficiency? Explain.

Yes. This confidence interval suggests that the patient may still have a calcium deficiency.Yes. This confidence interval suggests that the patient no longer has a calcium deficiency.    No. This confidence interval suggests that the patient may still have a calcium deficiency.No. This confidence interval suggests that the patient no longer has a calcium deficiency.

 

10.) What percentage of hospitals provide at least some charity care? Based on a random sample of hospital reports from eastern states, the following information is obtained (units in percentage of hospitals providing at least some charity care):

56.5
56.4
53.4
66.4
59.0
64.7
70.1
64.7
53.5
78.2

Assume that the population of x values has an approximately normal distribution.

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean percentage x and the sample standard deviation s. (Round your answers to one decimal place.)

(b) Find a 90% confidence interval for the population average μ of the percentage of hospitals providing at least some charity care. (Round your answers to one decimal place.)

lower limit
 %

upper limit
 %