1.
A person’s blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean μ = 85 and standard deviation σ = 23. Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (Round your answers to four decimal places.)
(a) x is more than 60
(b) x is less than 110
(c) x is between 60 and 110
(d) x is greater than 125 (borderline diabetes starts at 125)
Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 4.8 millimeters (mm) and a standard deviation of 1.7 mm. For a randomly found shard, find the following probabilities. (Round your answers to four decimal places.)
(b) the thickness is more than 7.0 mm
(c) the thickness is between 3.0 mm and 7.0 mm
3.
Find z such that 4.4% of the standard normal curve lies to the right of z. (Round your answer to two decimal places.)
z =
4.
Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 26 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is 4 months, and the distribution of lifetimes is normal.
(a) If Accrotime guarantees a full refund on any defective watch for 2 years after purchase, what percentage of total production will the company expect to replace? (Round your answer to two decimal places.)
%
(b) If Accrotime does not want to make refunds on more than 12% of the watches it makes, how long should the guarantee period be (to the nearest month)?
months
5.
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 79 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean μ = 79 tons and standard deviation σ = 0.5 ton.
(a) What is the probability that one car chosen at random will have less than 78.5 tons of coal? (Round your answer to four decimal places.)
(b) What is the probability that 17 cars chosen at random will have a mean load weight x of less than 78.5 tons of coal? (Round your answer to four decimal places.)
(c) Suppose the weight of coal in one car was less than 78.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment?
- Yes
- No
Suppose the weight of coal in 17 cars selected at random had an average x of less than 78.5 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.
6.
What price do farmers get for their watermelon crops? In the third week of July, a random sample of 45 farming regions gave a sample mean of x = $6.88 per 100 pounds of watermelon. Assume that σ is known to be $1.94 per 100 pounds.
(a) Find a 90% confidence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop. What is the margin of error? (Round your answers to two decimal places.)
| lower limit | $ |
| upper limit | $ |
| margin of error | $ |
(b) Find the sample size necessary for a 90% confidence level with maximal error of estimate E = 0.37 for the mean price per 100 pounds of watermelon. (Round up to the nearest whole number.)
farming regions
(c) A farm brings 15 tons of watermelon to market. Find a 90% confidence interval for the population mean cash value of this crop. What is the margin of error? Hint: 1 ton is 2000 pounds. (Round your answers to two decimal places.)
| lower limit | $ |
| upper limit | $ |
| margin of error | $ |
7.
How much do wild mountain lions weigh? Adult wild mountain lions (18 months or older) captured and released for the first time in the San Andres Mountains gave the following weights (pounds):
74, 108, 128, 121, 60,64
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 weight x and sample standard deviation s. (Round your answers to one decimal place.)
X= lb
S= lb
(b) Find a 75% confidence interval for the population average weight μ of all adult mountain lions in the specified region. (Round your answers to one decimal place.)
Lower limit= lb
Upper limit= lb
8.
Bill Alther is a zoologist who studies Anna’s hummingbird (Calypte anna).† Suppose that in a remote part of the Grand Canyon, a random sample of six of these birds was caught, weighed, and released. The weights (in grams) were as follows.
| 3.7 | 2.9 | 3.8 | 4.2 | 4.8 | 3.1 |
The sample mean is x = 3.75 grams. Let x be a random variable representing weights of hummingbirds in this part of the Grand Canyon. We assume that x has a normal distribution and σ = 0.74 gram. Suppose it is known that for the population of all Anna’s hummingbirds, the mean weight is μ = 4.50 grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.50 grams? Use α = 0.10.
- Find (or estimate) the P-value. (Round your answer to four decimal places.)
9.
Nationally, about 11% of the total U.S. wheat crop is destroyed each year by hail.† An insurance company is studying wheat hail damage claims in a county in Colorado. A random sample of 16 claims in the county reported the percentage of their wheat lost to hail.
| 16 | 10 | 8 | 11 | 11 | 21 | 16 | 10 |
| 7 | 10 | 26 | 18 | 12 | 7 | 13 | 5 |
The sample mean is x = 12.6%. Let x be a random variable that represents the percentage of wheat crop in that county lost to hail. Assume that x has a normal distribution and σ = 5.0%. Do these data indicate that the percentage of wheat crop lost to hail in that county is different (either way) from the national mean of 11%? Use α = 0.01.
(a) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.
- The Student’s t, since nis large with unknown σ.
- The standard normal, since we assume that xhas a normal distribution with known σ.
- The Student’s t, since we assume that xhas a normal distribution with known σ.
- The standard normal, since we assume that xhas a normal distribution with unknown σ.
What is the value of the sample test statistic? (Round your answer to two decimal places.)
(b) Find (or estimate) the P-value. (Round your answer to four decimal places.)
10.
Let x be a random variable that represents the pH of arterial plasma (i.e., acidity of the blood). For healthy adults, the mean of the x distribution is μ = 7.4.† A new drug for arthritis has been developed. However, it is thought that this drug may change blood pH. A random sample of 31 patients with arthritis took the drug for 3 months. Blood tests showed that x = 8.7 with sample standard deviation s = 2.9. Use a 5% level of significance to test the claim that the drug has changed (either way) the mean pH level of the blood.
(a) What is the value of the sample test statistic? (Round your answer to three decimal places.)
(b) Estimate the P-value.
- P-value > 0.500
- 250 < P-value < 0.500
- 100 < P-value < 0.250
- 050 < P-value < 0.100
- 010 < P-value < 0.050
- P-value < 0.010
11.
In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.
Ocean fishing for billfish is very popular in the Cozumel region of Mexico. In the Cozumel region about 40% of strikes (while trolling) resulted in a catch. Suppose that on a given day a fleet of fishing boats got a total of 23 strikes. Find the following probabilities. (Round your answers to four decimal places.)
(a) 12 or fewer fish were caught
(b) 5 or more fish were caught
(c) between 5 and 12 fish were caught
12.
In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.
It is estimated that 3.6% of the general population will live past their 90th birthday. In a graduating class of 751 high school seniors, find the following probabilities. (Round your answers to four decimal places.)
(a) 15 or more will live beyond their 90th birthday
(b) 30 or more will live beyond their 90th birthday
(c) between 25 and 35 will live beyond their 90th birthday
(d) more than 40 will live beyond their 90th birthday
13.
Suppose the heights of 18-year-old men are approximately normally distributed, with mean 67 inches and standard deviation 6 inches.
(a) What is the probability that an 18-year-old man selected at random is between 66 and 68 inches tall? (Round your answer to four decimal places.)
(b) If a random sample of ten 18-year-old men is selected, what is the probability that the mean height x is between 66 and 68 inches? (Round your answer to four decimal places.)
(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?
- The probability in part (b) is much higher because the mean is smaller for the x
- The probability in part (b) is much higher because the mean is larger for the x
- The probability in part (b) is much lower because the standard deviation is smaller for the x
- The probability in part (b) is much higher because the standard deviation is larger for the x
- The probability in part (b) is much higher because the standard deviation is smaller for the x
14.
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 = 6700 and estimated standard deviation σ = 2850. 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 approximately normal with μx = 6700 and σx = 1425.00.
- The probability distribution of x is not normal.
- The probability distribution of x is approximately normal with μx = 6700 and σx = 2015.25.
- The probability distribution of x is approximately normal with μx = 6700 and σx = 2850.
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 increased as n
- The probabilities decreased as n
- The probabilities stayed the same as n
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 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 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 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.
15.
Sketch the area under the standard normal curve over the indicated interval and find the specified area. (Round your answer to four decimal places.)
The area between
z = −2.26 and z = 1.45 is
The area between
z = −2.32 and z = −1.88 is
16.
Porphyrin is a pigment in blood protoplasm and other body fluids that is significant in body energy and storage. Let x be a random variable that represents the number of milligrams of porphyrin per deciliter of blood. In healthy circles, x is approximately normally distributed with mean μ = 40 and standard deviation σ = 11. Find the following probabilities. (Round your answers to four decimal places.)
(a) x is less than 60
(b) x is greater than 16
(c) x is between 16 and 60
(d) x is more than 60 (This may indicate an infection, anemia, or another type of illness.)
17.
Police response time to an emergency call is the difference between the time the call is first received by the dispatcher and the time a patrol car radios that it has arrived at the scene. Over a long period of time, it has been determined that the police response time has a normal distribution with a mean of 8.5 minutes and a standard deviation of 1.8 minutes. For a randomly received emergency call, find the following probabilities. (Round your answers to four decimal places.)
(a) the response time is between 5 and 10 minutes
(b) the response time is less than 5 minutes
(c) the response time is more than 10 minutes
18.
Find z such that 8.1% of the standard normal curve lies to the right of z. (Round your answer to two decimal places.)
z =
Sketch the area described.
19.
Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 26months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is 6 months, and the distribution of lifetimes is normal.
(a) If Accrotime guarantees a full refund on any defective watch for 2 years after purchase, what percentage of total production will the company expect to replace? (Round your answer to two decimal places.)
%
(b) If Accrotime does not want to make refunds on more than 16% of the watches it makes, how long should the guarantee period be (to the nearest month)?
months
