Use the following context for the questions 1 thru 5

It is given that 60% of the employees at a large company are women. A simple random sample of 15 employees is selected from this company.

We are applying the binomial distribution in this situation.

1. What is the mean of this binomial distribution?

2. What is the standard deviation of this binomial distribution?

3. Find the probability that the number of women in this sample will be exactly 8.

4. Find the probability that the number of women in this sample will  8 or less.

5. Find the probability that the number of women in this sample will be more than 8.

Consider the following context for #6-8

A university has determined that the probability that a new student (with confirmed dorm assignment) will actually take the dorm space is 0.87. The university has 800 spaces for new students.

6. How many students are expected to take the dorm space if the number of confirmed assignments is 800?

7. The university confirms the dorm assignments for 870 students, use the binomial distribution to find the probability that the number of students actually taking the dorm space will be 800 or less.

8. Is the university taking too high a risk in sending confirmation to870 students under the assumption that p=0.87 and that all students make decisions independently?

Consider the following statement for the questions 9-15

Given that the lengths of human pregnancies are normally distributed with a mean of 266 days and standard deviation 16 days.

9. Find the probability that a human pregnancy will be less than 260 days long.

10. Find the probability that a human pregnancy will be less than 280 days long.

11. Find the probability that a human pregnancy will be between 260 days and 280 days long.

12. Find the 90th percentile of the length of human pregnancies.

13. Explain in one sentence, what does the answer for 90th percentile mean in this context.

13. Find the probability that a human pregnancy will be longer than 280 days.

14. An insurance company has 10 cases of pregnancies among its insured people. Use the binomial distribution that no more than 2 of these will be longer than 280 days.

Consider the following for the questions 16-20,

An insurance company has determined that the mean of the claim payments on collusions in a certain region is $1300 and standard deviation $2400.

The company has 100 claims coming up and it would like to treat these 100 as a simple random sample from the population with mean $1300 and standard deviation $2400.

16. What is the mean of the means of simple random samples of size 100 from this population?

17. What is the standard deviation of the means of simple random samples of size 100 from this population?

18. Does the reference population in this situation have a normal distribution, give a reason?

19. Why can we still treat the distribution of sample means for simple random samples of size 100 as approximately normal?

20. Find the probability that the total claim of a simple random sample of 100 claims will be more than $200,000.