1. Assume that the life of an electrical bulb is normally distributed with a mean of 1000 hours and a standard deviation of 170 hours. Find the following probability.
a. What is the probability that the bulb will last for more than 1150 hours?
b. What is the probability that the bulb will last for less than 700 hours?
c. What is the probability that the bulb will last between 750 and 1200 hours?
2. A hospital's patient account division has compiled data on the age of accounts receivable. The data collected indicate that the age of the accounts follows a normal distribution, with a mean of 19 and a standard deviation of 10 days.
a) What proportion of the accounts are between 18 and 30 days old?
b) The hospital administrator is interested in sending reminder letters to the oldest 25% accounts. How many days should an account be before a reminder letter is sent?
3. Southern HealthCare, a multi-hospital health care system, has a significant number of duplicate patient registrations in its database of 2.2 million patients. Duplicate registrations create potential risk for Southern as information gathered during different patient encounters such as allergic reactions to medications, emergency contacts, insurance information and powers of attorney can be scattered throughout the database and not readily available to clinical staff treating the patients. Additionally, staff in Medical Records, Laboratory, Radiology and Pharmacy spend hundreds of hours annually researching and merging duplicate records.
Duplicates are created when a patient arrives at a facility and their existing registration is not located in the computer system. Not locating existing registrations can be attributed to two factors: personnel and the hospital computer system. In this problem, we focus on the distribution of the weekly number of duplicates. It was discovered that the number of duplicates per week follows a normal distribution, with a mean of 47 and a standard deviation of 8. Answer the following questions from the management.
a) If the number of duplicates during a week exceeds 58, at the end of that week a special meeting will be held. Find the percentage of weeks during which special meetings will be held.
b) What is the 60th percentile?
c) What is the 85th percentile?
d) What is the first quartile (in Minitab, it is denoted Q1), i.e., the 25th percentile?
4. Suppose
that a government agency wants to estimate the average number of kids per
family in
5. A researcher is interested in estimating the average purchase amount for convenience stores. To do so, she randomly sampled 32 purchases from several convenience stores and tabulated the amounts to the nearest dollar. Use the following data to construct a 95% confidence interval for the population average amount of purchase:
|
$2 |
$11 |
$8 |
$7 |
|
5 |
4 |
2 |
1 |
|
14 |
7 |
9 |
3 |
|
4 |
3 |
3 |
8 |
|
9 |
3 |
1 |
6 |
|
10 |
8 |
5 |
4 |
|
7 |
5 |
3 |
9 |
|
6 |
4 |
7 |
12 |
6. The National Automobile Dealers Association collects data on sales numbers, prices, and usage of both new and used automobiles. One statistic of interest is the percentage of automobiles that are still on the road after 10 years ( U.S. News & World Report, September 9, 1991). A random sample of 1000 cars were selected in 1991.
a) Assume that 25% of automobiles sampled in 1991 were still on the road after 10 years. Provide a 95% confidence interval for the population proportion. What is your interpretation of this confidence interval?
b) How large a sample should be taken if we want to be 95% confident that the sample percentage is within 2.5% of the actual percentage of automobile that are still on the road after 10 years?
7. In
1984, there were about 60 million cards in the
a) What are the population and the unit?
b) Given that "little is known about the shape of the distribution of billing amounts", can you compute the probability that the sample mean exceeds $940? If yes, compute it. If no, why not?
8. A
real estate company wishes to estimate, with 95 percent confidence, the
difference between the average rental prices of two-bedroom apartments in two
cities,
a) From the given information, can you tell us which city has a higher average rental price at 95% confidence? Explain.
b) If the answer in a) is inconclusive, what should you do in order to obtain a conclusive solution at the same confidence level?
9. Schumacher is a supplier of high purity chemicals to the semiconductor industry. Most of Schumacher’s products are used for the manufacture of integrated circuit chips. Because of the extremely high complexity and numerous process steps, Schumacher must always provide consistent product to its customers. A project has been initiated to study incoming raw material quality and how it effects the final product. In this problem we focus on two major materials used by the company, called Component A and B. Two random samples, one for component A, the other for Component B were collected by the analytical department. The result is shown in the following table. The reason for the small number of data points is that previously this product was manufactured in R&D Department on small scale. Large batch processing has just been started. Since sample sizes are not large enough, we will assume that the distribution of putity is indeed normal.
|
|
Purity (%) |
|
|
|
Comp. A |
Comp. B |
|
1 |
99.46 |
98.8 |
|
2 |
99.47 |
99.45 |
|
3 |
99.54 |
99.45 |
|
4 |
99.64 |
99.53 |
|
5 |
99.62 |
99.53 |
|
6 |
99.75 |
99.63 |
|
7 |
99.68 |
99.61 |
|
8 |
99.69 |
99.12 |
|
9 |
99.93 |
99.54 |
|
10 |
99.51 |
99.12 |
|
11 |
99.67 |
99.54 |
|
12 |
99.62 |
99.51 |
|
13 |
99.62 |
99.52 |
|
14 |
99.67 |
99.17 |
|
15 |
99.63 |
|
|
16 |
99.68 |
|
a) Use Minitab/Excel to
construct separate 99% confidence intervals of the average purity for Component
A and B.
b)
Use
Minitab/Excel to construct the 99% confidence interval for the difference of
the two population means. Can you
conclude that Component A has higher average purity or vice versa?
10. The city manager wants to compare the average rental prices of the coastal and inland areas of her city. The rental prices of randomly selected coastal 2-bedroom apartments and inland 2-bedroom apartments in her city are obtained and stored in two columns named COAST and INLAND in a Minitab file “Rental.mtw”.
a) Construct a 95% confidence interval of the difference of the average rental prices of the two areas using the formula in the text. You can use Minitab to compute certain statistics needed for your calculation. At 95% confidence can you tell the manager what area has higher average rental price? By how much?
b) Use Minitab to construct the 95% confidence interval of the difference of the average rental prices of the two areas. Is the result the same as the one calculated in a)? If it were different, would it change your answers to those questions listed in a)?
11. The
Vallecitos Water District is a public agency that provides water and wastewater
treatment and disposal in the greater
Forty seven (47) single family residences were randomly selected. Their water consumptions in the three months of 1989 and 1995 were entered into two columns as shown below.
|
Family |
Total-89 |
Total-95 |
|
1 |
59 |
74 |
|
2 |
47 |
48 |
|
3 |
14 |
64 |
|
4 |
49 |
61 |
|
5 |
35 |
12 |
|
6 |
56 |
69 |
|
7 |
158 |
135 |
|
8 |
41 |
43 |
|
. |
. |
. |
|
. |
. |
. |
|
. |
. |
. |
The unit is 100
cubic feet, which is equal to 748 gallons.
For example, the first family in the sample consumed 5,900 cubic feet of
water in 1989 for the months of July, August, and September, and 7,400 cubic
feet in the same period of 1995. The
second family consumed 4,700 in 1989 in the same period, and so forth. The data file is called Water-usage.mtw.
Answer the
following questions.
a) Find the mean, median, and standard deviation of water consumptions for both years.
b) Does water consumption follow a normal distribution? Use Minitab/Excel to draw frequency distributions of the two variables to answer this question. Can you explain why the distribution has its form?
c) Find the 95% confidence intervals for the average water consumptions for the two years.
d) Find the 95% confidence interval for the difference of the average water consumptions of the two years. Do you see a clear difference of the two years?