The Selayang Municipal Council (MPS) reported the following information for a sample of 250 customers on the number of hours their cars are parked and the amount they are charged:
Number of hours | Frequency | Amount charged (RM) |
1 | 20 | 3.00 |
2 | 38 | 6.00 |
3 | 53 | 9.00 |
4 | 45 | 12.00 |
5 | 40 | 14.00 |
6 | 13 | 16.00 |
7 | 5 | 18.00 |
8 | 36 | 20.00 |
250 |
a. Convert this information on the number of hours parked to a probability distribution. Is this a discrete or continuous probability distribution?
b. Find the mean and the standard deviation of the number of hours cars are parked. How long is a typical customer parked?
c. Find the mean and standard deviation of the amount charged.
Example 2
The random variable, and each corresponding probability is given below.
1 | 2 | 4 | 10 | |
0.3 | 0.2 | 0.2 | ? |
Answer the following:
a. Find
b. Find
Example 3
If the probability of a sunny day is 0.4, find the expected number of sunny days in a week, and the standard deviation.
Example 4
Besides physical outlets in major cities in Malaysia, Nafa Electronics implemented e-commerce last year in order to sell their products online. Recent study shows that 90 percent of their products in the same city are delivered within two days. 15 products are randomly sent to different locations:
a. What is the probability that all 15 products will be delivered within 2 days?
b. What is the probability that exactly 10 products will be delivered within 2 days?
c. Find the mean number of products that will be delivered within 2 days.
d. Compute the variance and standard deviation of the number of products that will be delivered within 2 days.
Example 5
A telemarketer makes six phone calls per hour and is able to make a sale on 30 per cent of these contacts. During the next two hours, find
a. The probability of making exactly four sales
b. The probability of making no sales
c. The probability of making exactly two sales
d. The mean number of sales in the two-hour period
Example 6
An important part of the customer service of a telephone company relate to the speed with which troubles in residential service can be repaired. Suppose past data indicate that 70% of troubles in the service can be repaired on the same day. For the first five troubles reported on a given day, what is the probability that
a. All five will be repaired on the same day?
b. At least three will be repaired on the same day?
c. Fewer than two will be repaired on the same day?
d. What are the mean and standard deviation of this probability distribution?
Example 7
At the Express House Delivery Service, providing high-quality service to customers is the top priority of the management. The company guarantees a refund of all charges if a package does not arrive at the specified time. It is known from past data that despite all efforts, 2 percent of the packages mailed through this company do not arrive at their destinations within the specified time. Suppose a corporation mails 10 packages through Express House Delivery Service on a certain day.
a. Find the probability that exactly one of these packages will not arrive at its destination within the specified time.
b. Find the probability that at most one of these packages will not arrive at its destination within the specified time.
Example 8
On average, there are 3.2 defects in a sheet of rolled steel. Assume that the number of defects follows a Poisson distribution
a. What is the probability of having exactly 3 defects in a roll?
b. What is the probability of having more than 3 defects in a roll?
Example 9
A book containing 500 pages has 750 misprints.
a. What is the average number of misprints per page?
b. Find the probability that on page 427, it contains
i. No misprints
ii. Exactly 4 misprints
c. More than the average number of misprints
d. Find the probability that pages 450 and 451 will contain no misprints.
Example 10
The National Aeronautics and Space Administration (NASA) has experienced two disasters. The Challenger exploded over the Atlantic Ocean in 1986 and the Columbia exploded over East Texas in 2003. There have been a total of 123 space missions. Assume failures continue to occur at the same rate and consider the next 25 missions,
a. What is the probability of exactly two failures?
b. What is the probability of no failure?
Example 11
The mean of a normal probability distribution is 500 and the standard deviation is 10.
a. About 68% of the observations lie between what values?
b. About 95% of the observations lie between what values?
c. Virtually all of the observations lie between what values?
Example 12
A recent national survey concluded that the typical person consumes 48 ounces of water per day. Assume daily water consumption follows a normal probability distribution with a standard deviation of 12.8 ounces.
a. What is the z value for a person who consumes 64 ounces of water per day? How does this person compare to the national average?
b. What is the z value for a person who consumes 32 ounces of water per day? How does this person compare to the national average?
Example 13
General Hospital’s patient account division has compiled data on the age of accounts receivables. The data collected indicate that the age of the accounts follow a normal distribution with mean 28 days and standard deviation 8 days.
a. What is the probability that the accounts are between 20 and 40 days old?
b. What is the probability that the accounts are less than 30 days old?
Example 14
A building contractor claims he can renovate a 200 sq. ft. kitchen and dining room in 40 work hours, plus or minus 5 hours. The work includes plumbing, electrical, cabinets, flooring, new appliance installation and painting. Assume from past experience that times to complete similar projects are normally distributed with mean and standard deviation as estimated above.
a. What is the likelihood the project will be completed in less than 35 hours?
b. What is the likelihood the project will be completed in between 28 and 32 hours?
c. What is the likelihood the project will be completed in between 43 and 48 hours?
d. What is the likelihood that the project will be completed in between 38 and 45 hours?
e. What is the likelihood the project will be completed in more than 42 hours?
Example 15
Among US cities with a population of more than 250,000 the mean one-way commute to work is 24.3 minutes. The longest one-way travel time is New York City, where the mean time is 38.3 minutes. Assume the distribution of travel times in NYC follows the normal distribution and the standard deviation is 7.5 minutes.
a. What percent of the NYC commutes are for less than 30 minutes?
b. What percent are between 30 and 35 minutes?
c. What percent are between 30 and 40 minutes?
Example 16
For the most recent year available the mean annual cost to attend a private university in the US was $26,889. Assume the distribution of annual costs follows a normal distribution and the standard deviation is $4,500. Ninety-five percent of all students at private universities pay
less than what amount?
Example 17
The U.S Environmental Protection Agency publishes figures on solid waste generation in the United States. One year, the average number of wastes generated per person per day was 3.58 pounds. Suppose the daily amount of waste generated per person is normally distributed, with a standard deviation of 1.04 pounds. Of the daily amounts of waste generated per person, 67.72 per cent would be
greater than what amount?
Example 18
In one county, the conviction rate for speeding is 85%. Estimate the probability that of the next 100 speeding summonses issued, there will be at least 90 convictions.
Example 19
Cool King advertises it can install a new air-conditioner in 30 minutes or less. However, in a recently conducted study by the headquarters, it was found that 20 percent of the air-conditioners were not installed in 30 minutes or less. One of the branches installed 50 air conditioners last month. If the study is correct,
a. How many of the installations at the branch would you expect to take more than 30 minutes?
b. What is the likelihood that fewer than eight installations took more than 30 minutes?
c. What is the likelihood that eight or fewer took more than 30 minutes?
d. What is the likelihood that exactly eight of the 50 installations took more than 30 minutes?
Example 20
IIUM has announced that it will admit 2046 students for its upcoming September intake. However, the university has also announced that it only has mahallah room spots for 1655 first year students. If there is an 80% chance that an admitted student will decide to accept the offer and attend IIUM, what is the probability that IIUM will not have enough mahallah room spots for the 1st year students?
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