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# Probability and Standard Deviation.

#1) The U.S. Bureau of Labor Statistics collected data on the occupations of workers 25 to 64 years old. The following table shows the number of male and female workers (in millions) in each occupation category (Statistical Abstract of the United States: 2002)
Occupation Male Female
Managerial/Professional 19079 19021
Service 4977 7947
Precision Production 11682 1138
Operations/Fabrications/Labor 10576 3482
Farming/Forestry/Fishing 1838 514

a. Develop a joint probability table
b. Determine the probability of a female worker being a manager or professional.
c. Determine the probability of a male worker being in precision production.
d. Determine if occupation is independent of gender. Justify the answer with a probability calculation.

#2) In an article about investment growth, Money magazine reported that drug stocks show powerful long-term trends and offer investors unparalleled potential for strong and steady gains. The federal Health Care Financing Administration supports this conclusion through its forecast that annual prescription drug expenditures will reach \$366 billion by 2010, up from \$117 billion in 2000. Many individuals age 65 and older rely heavily on prescription drugs. For this group, 82% take prescription drugs regularly, 55% take three or more prescriptions regularly, and 40% currently use five or more prescriptions prescriptions. In contrast, 49% of people under age 65 take prescriptions regularly, with 17% taking three or more prescriptions regularly and 28% using five or more prescriptions (Money, September 2001) The U.S. Census Bureau reports that of the 281,421,906 people in the United States, 34,991,753 are age 65 years and older (U.S. Census Bureau, Census 2000).
a. Determine the probability that a person in the United States is age 65 or older.
b. Determine the probability that a person takes prescription drugs regularly.
c. Determine the probability that a person is age 65 or older and takes five or more prescriptions.
d. Given that a person uses five or more prescriptions, determine the probability that the person is age 65 or older.

#3) Fifty percent of Americans believed the country was in a recession, even though technically had not shown two straight quarters of negative growth (Business Week, July 30, 2001). For a sample of 20 Americans, make the following calculations.
a. Determine the probability that exactly 12 people believed the country was in a recession.
b. Determine the probability that no more than five people believed the country was in a recession.
c. How many people would you expect to say the country was in a recession.
d. Determine the variance and standard deviation of the number of people who believed the country was in a recession.

#4) Is lack of sleep causing traffic fatalities? A study conducted under the auspices of the National Highway Traffic Safety Administration found that the average number of fatal crashes caused by drowsy drivers each year was 1550 (Business Week, January 26, 2004). Assume the annual number of fatal crashes per year is normally distributed with a standard deviation of 300.
a. Determine the probability of fewer than 1000 fatal crashes in a year.
b. Determine the probability the number of fatal crashes will be between 1000 and 2000 for a year.
c. For a year to be in the upper 5% with respect to the number of fatal crashes, determine how many fatal crashes would have to occur.

#### Solution Preview

#1)
The U.S. Bureau of Labor Statistics collected data on the occupations of workers 25 to 64 years old. The following table shows the number of male and female workers (in millions) in each occupation category (Statistical Abstract of the United States: 2002)
Occupation Male Female
Managerial/Professional 19079 19021
Service 4977 7947
Precision Production 11682 1138
Operations/Fabrications/Labor 10576 3482
Farming/Forestry/Fishing 1838 514

Male Female Total
Managerial / Professional 19,079 19,021 38,100
Tech / Sales / Administrative 11,079 19,315 30,394
Service 4,977 7,947 12,924
Precision Production 11,682 1,138 12,820
Operation / Fabrication labor 10,576 3,482 14,058
Farming / Forestry / Fishing 1,838 514 2,352
Total 59,231 51,417 110,648

a. Develop a joint probability table

We develop a joint probability table by dividing all the numbers by the total number= 110,648

Male Female Total
Managerial / Professional 0.1724 0.1719 0.3443
=19,079 / 110,648 =19,021 / 110,648 =0.1724 + 0.1719
Tech / Sales / Administrative 0.1001 0.1746 0.2747
=11,079 / 110,648 =19,315 / 110,648 =0.1001 + 0.1746
Service 0.0450 0.0718 0.1168
=4,977 / 110,648 =7,947 / 110,648 =0.045 + 0.0718
Precision Production 0.1056 0.0103 0.1159
=11,682 / 110,648 =1,138 / 110,648 =0.1056 + 0.0103
Operation / Fabrication labor 0.0956 0.0315 0.1271
=10,576 / 110,648 =3,482 / 110,648 =0.0956 + 0.0315
Farming / Forestry / Fishing 0.0166 0.0046 0.0212
=1,838 / 110,648 =514 / 110,648 =0.0166 + 0.0046
Total 0.5353 0.4647 1.0000

b. Determine the probability of a female worker being a manager or professional.

Probability of female= 0.4647
Probability of female and being professional or manager= 0.1719
Therefore probability that a worker is manager or professional given that she is female= 0.3699 =0.1719 / 0.4647

c. Determine the probability of a male worker being in precision production.

Probability of male= 0.5353
Probability of male and being in precision production= 0.1056
Therefore probability that a worker is in precision production given that he is male= 0.1973 =0.1056 / 0.5353

d. Determine if occupation is independent of gender. Justify the answer with a probability calculation.

If events are independent , P ...

#### Solution Summary

Answers 4 questions on calculation of probability, standard deviation etc.

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