Q1. 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?
Q2. 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?
Q3. 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
|
?
|
|
Q4. 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 above question. 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
|
?
|
|
Q5. Tastes in music:
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.)