# Probability and Statistics

CH3 Problem 50

The mean income of a group of sample observations is $500; the standard deviation is $40. According to Chebyshev's theorem, at least what percent of the incomes will lie between $400 and $600?

CH3 Problem 62

The Citizens Banking Company is studying the number of times the ATM located in a Loblaws Supermarket at the foot of Market Street is used per day. Following are the numbers of times the machine was used over each of the last 30 days. Determine the mean number of times the machine was used per day.

83 64 84 76 84 54 75 59 70 61

63 80 84 73 68 52 65 90 52 77

95 36 78 61 59 84 95 47 87 60

CH5 Problem 28

Three defective electric toothbrushes were accidentally shipped to a drugstore by Cleanbrush Products along with 17 nondefective ones. What is the probability the first two electric toothbrushes sold will be returned to the drugstore because they are defective?

CH5 problem 78

The state of Maryland has license plates with three numbers followed by three letters. How many different license plates are possible?

CH7 problem 20

The mean starting salary for college graduates in the spring of 2004 was $36,280. Assume that the distribution of starting salaries follows the normal distribution with a standard deviation of $3,300. What percent of the graduates have starting salaries:

a. Between $35,000 and $40,000?

b. More than $45,000?

c. Between $40,000 and $45,000?

CH7 problem 40

It is estimated that 10 percent of those taking the quantitative methods portion of the CPA examination fail that section. Sixty students are taking the exam this Saturday.

a. How many would you expect to fail? What is the standard deviation?

b. What is the probability that exactly two students will fail?

c. What is the probability at least two students will fail?

https://brainmass.com/statistics/probability/probability-and-statistics-121376

#### Solution Preview

CH3 Problem 50

The mean income of a group of sample observations is $500; the standard deviation is $40. According to Chebyshev's theorem, at least what percent of the incomes will lie between $400 and $600?

This theorem says that:

For any population or sample, at least (1 - (1 / k)2) of the observations in the data set fall within k standard deviations of the mean, where k  1.

Here, we want to know what percentage of the incomes will fall within $100 of the mean. Put this in terms of a multiple of standard deviations (i.e. find k):

100 = 40k

k = 100/40 = 2.5

Now, we know that we are interested in the percent of incomes that fall within 2.5 standard deviations of the mean. Use the theorem to solve for the percent of incomes in this range:

(1 - (1 / k)2) = (1 - (1 / 2.5)2) = (1 - (0.4)2) = (1 - 0.16) = 0.84

At least 84% of the incomes will lie between $400 and $600.

CH3 Problem 62

The Citizens Banking Company is studying the number of times the ATM located in a Loblaws Supermarket at the foot of Market Street is used per day. Following are the numbers of times the machine was used over each of the last 30 days. Determine the mean number of times the machine was used per day.

83 64 84 76 84 54 75 59 70 61

63 80 84 73 68 52 65 90 ...

#### Solution Summary

The solution consists of complete answers and explanations to six questions. These questions are on:

-- Chebyshev's Theorem

-- mean and standard deviation

-- probability and combinations

-- normal distribution

-- binomial distribution

Statistics and Probability in Computing

1) When sending data over the internet there is a certain probability that a message will be corrupted. One way to improve the reliability of getting messages through is to use a Hamming Code. This involves sending extra data that can be used to check the main message. For example a 7 bit Hamming Code contains 4 bits of message data and 3 check bits. If only one of the bits is in error at the receiving end then mathematical techniques can be used to determine which one it is and apply a correction. Assume that you have a network connection for which the probability that an individual bit will get through without error is 0.66. What is the increase in the probability that a 4 bit message will get through if a 7 bit Hamming code is used instead of just sending the 4 bits? (i.e what is P(7 bits with 0 or 1 error) - P(4 bits with no error)?

2) Q Computers has invented quantum computers. Each computer contains an exotic sub-atomic particle. Unfortunately this particle decays in the same manner as all radioactive particles. Therefore an average quantum computer only lasts for 22 months. The University has purchased one of these computers and Professor Squiggle wants to use it for 7 months. When he tries to book it he finds that it is already booked out for the first 8 months. So he books it for the next 7 months. What is the probability that the computer will fail during the time that professor Squiggle is using it (not before and not after)?

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