1. Length of pregnancies
The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days.
a) What percent of pregnancies last fewer than 240 days (that's about 8 months)?
b) What percent of pregnancies last between 240 and 270 days (roughly between 8 and 9 months)?
c) How long do the longest 25% of pregnancies last?
2. deciles of Normal distributions
The deciles of any distribution are the 10th, 20th..., 90th percentiles, respectively.
a) What are the first and last deciles of the standard Normal distribution?
b) The weights of 9-ounce potato chip bags are approximately Normal with mean 9.25 ounces and standard deviation 0.15 ounce. What are the first and last deciles of this distribution?
3. Life Insurance
A life insurance company sells a term insurance policy to a 21-year-old male that pays $100,000 if that insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $250 each year as payment for the insurance. The amount X that the company earns on this policy is $250 per year., less the $100,000 that it must pay if the insured dies. The distribution of X is shown below. Fill in the missing probability in the table and calculate the mean earnings mx.
Age at Death (years)
|
Age of Death
|
|
|
21
|
22
|
23
|
24
|
25
|
>26
|
|
|
Earnings X
Probability
|
($99,750)
|
($99,550)
|
($99,250)
|
($99,000)
|
($98,750)
|
$1,250
|
|
|
0.00183
|
0.00186
|
0.00189
|
0.00191
|
0.00193
|
?
|
|
4. More about life insurance
It would be quite risky for you to insure the life of a 21-year-old friend under the terms of Exercise 4.151. There is a high probability that your friend would live and you would gain $1,250 in premiums. But if he were to die, you would lose almost $100,000. Explain carefully why selling insurance is not risky for an insurance company that insure many thousands of 21-year-old men?
|
Age of Death
|
|
|
21
|
22
|
23
|
24
|
25
|
>26
|
|
|
Earnings X
Probability
|
($99,750)
|
($99,550)
|
($99,250)
|
($99,000)
|
($98,750)
|
$1,250
|
|
|
0.00183
|
0.00186
|
0.00189
|
0.00191
|
0.00193
|
?
|
|
5 Tastes in music (pp.303)
Musical styles other than rock and pop are becoming more popular. A survey of college students finds that 40% like country music, 30% like gospel music, and 10% like both.
a) What is the conditional probability that a student likes gospel music if we know that he or she likes country music?
b) What is the conditional probability that a student who does not like country music likes gospel music? (A Venn diagram may help you.)