# Arithmetic mean, median and probablity

1. A property of concern for any food company that uses a high-speed carton-filling machine to package juice is the weight of the food product in the individual cartons. If the cartons are under filled, two problems arise. First, customers may not have enough product for their needs. Second, the company may be in violation of the truth-in-labeling laws. In this example, the label weight on the package indicates that, on average, there are 2.5 ounces of product in a carton. If the average amount of product in a carton exceeds the label weight, the company is giving the product for free. Getting an exact amount of product in a carton is problematic because of variation in the temperature and humidity inside the factory, differences in the density of the product, and the extremely fast filling operation of the machine (approximately 450 cartons per minute). The following table provides the weight in ounces of a sample of 60 cartons produced in one hour by a single machine:

4.01 4.06 2.45 2.06 2.02 2.59 2.72 3.08 3.08 3.04

2.22 2.47 2.96 2.41 2.42 2.09 3.03 3.09 1.98 3.05

3.11 2.31 1.28 3.01 2.42 2.49 1.57 2.46 2.55 2.52

3.02 1.87 1.99 1.88 1.38 3.06 3.04 3.04 3.07 2.52

3.08 3.03 1.62 2.32 1.43 2.12 2.43 2.99 1.85 3.08

3.04 3.11 1.59 1.81 3.02 2.99 3.01 1.76 3.01 2.33

a. Compute the arithmetic mean and median.

b. Compute the first quartile and third quartile.

c. Compute the range, interquartile range, variance, standard deviation, and coefficient of variation.

d. Interpret the measures of central tendency within the context of this problem. Why should the company producing the bottles be concerned about the central tendency?

e. Interpret the measures of variation within the context of this problem. Why should the company producing the bottles be concerned about variation?

https://brainmass.com/statistics/probability-theory/arithmetic-mean-median-and-probablity-569799

#### Solution Summary

This solutions shows how to calculate descriptive statistics in Excel and how to find probabilities of events using basic principles of probability.

Statistics : Distributions, Probability, Means, Quartiles and Five-Number Summary

1. The array below shows the number of days required for approval for

a random selection of small business loan applications:

56 21 38 17 15 67 45 48 60 56 22 53 37 17

50 92 39 16 59 73 17 31 28 51 74 42 31

What is the arithmetic mean for these observations? (Round the

calculation to two decimal places.)

a. 42.00

b. 42.78

c. 41.25

d. None of the above

2. Using the array in #1 above, calculate the median of the

distribution, rounding to two decimal places.

a. 42.00

b. 42.78

c. 41.25

d. None of the above

3. Using the array in question 1 above, calculate the arithmetic mean

with the single most extreme outlier removed. Round the

calculation to two decimal places.

a. 40.50

b. 40.89

c. 41.92

d. 43.85

4. The following is a five-number summary calculated from

observations of the percentage of calories that come from fat in 23

items advertised as "reduced" fat from a sample of fast food chain

restaurants. Form the box-and-whisker plot, select the phrase

below which bests describes the shape of the distribution.

__________________________________

| 15 | 20 | 22 | 26 | 30 |

|____ |______ |______ |_______|_______|

a. Skewed slightly to the left

b. Skewed slightly to the right

c. Symmetrical distribution

d. There is not enough information given to determine the shape

5. The following data represents the processing times required to pack

20 randomly selected computer hardware orders for mailing:

5.62 16.25 11.46 8.45 5.41 11.62 7.5 4.42 7.58 7.54

5.29 10.92 21.62 8.58 11.42 7.29 7.96 10.5 9.29 8.92

What is the third quartile?

a. 10.92

b. 11.05

c. 11.42

d. 11.46

6. What is the standard deviation of the distribution of the

observations in the array shown in question 11 above?

a. 0.894

b. 3.998

c. 4.199

d. 15.981

7. What is the coefficient of variation for the observations in the array

shown in question 11 above?

a. 39.97 %

b. 44.15%

c. 42.61%

d. 46.94 %

8. The employees of a company were surveyed on questions regarding

their educational background and marital status. Of the 600

employees, 400 had college degrees, 100 were single, and 60 were

single college graduates. The probability that an employee of the

company is married and has a college degree is:

a. 40/600

b. 340/600

c. 400/600

d. 500/600

9. An advertising executive is studying TV viewing habits of married

men and women during prime time hours. On the basis of past

viewing records, the executive has determined that during prime

time, husbands are watching TV 60% of the time that their wives

are also watching TV. When the husband is not watching TV, 30%

of the time his wife is watching TV. What is the probability that, if

the wife is watching TV, her husband is also watching TV?

a. 0.12

b. 0.24

c. 0.36

d. 0.67

10. Mothers Against Drunk Driving is a very visible group whose main

focus is to educate the public about the harm caused by drunk

drivers. A study was recently done that emphasized the problem we

all face with drinking and driving. Four hundred accidents that

occurred on a Saturday night were analyzed. Two items noted were

the number of vehicles involved and whether alcohol played a role in

the accident. The numbers are shown below:

Number of vehicles Involved

Did alcohol play a role? 1 2 3 Totals

Yes 50 100 20 170

No 25 175 30 230 1

Totals 75 275 50 400

Referring to the table above, what proportion of accidents involved

more than one vehicle?

a. 50/400

b. 75/400

c. 275/400

d. 325/400

11. Referring to the table in question 20 above, what proportion of the

non-alcohol related accidents were multiple vehicle accidents?

a. 50/170

b. 120/170

c. 205/230

d. 25/230

12. A company has 2 machines that produce widgets. An older machine

produces 23% defective widgets, while the new machine produces

only 8% defective widgets. In addition, the new machine produces 3

times as many widgets as the older machine does. What is the

probability that a randomly chosen widget produced by the company

is defective?

a. .055

b. .0775

c. .1175

d. .256

13. A survey is taken among customers of a fast-food restaurant to

determine preference for hamburger or chicken. Of 200 respondents

selected, 75 were children and 125 were adults. 120 preferred

hamburger and 80 preferred chicken. 55 of the children preferred

hamburger. What is the probability that a randomly selected

individual is an adult?

a. 0.50

b. 0.625

c. 0.75

d. None of the above