Exercises. 1. A large population has mean 90 and standard deviation 8

1. A large population has mean 90 and standard deviation 8. Calculate and for a random sample of size

a) n =9 and b) n =36.

2. A large population has a standard deviation 10. What is the standard deviation of for a random sample of size

a) n =25; b) n =100; and c) n =400

3. Consider a large population with and

Assuming , find the mean and standard deviation of the sample

mean for the sample size of

a) 18; b) 90

4. A population of N =5000 has a . In each of the following cases which formula you use to calculate and why?

Using the appropriate formula, calculate for each of these cases.

a) n =300; b) n = 200; c) n = 500; d) n = 100.

5. For a population with and

a) For a sample selected from this population, and .

Find the sample size. Assume .

b) For a sample selected from this population, and . Find the sample size. Assume .

6. The following data show the number of automobiles owned in a population of five families:

2; 1; 0; 2; 3

a) List the ten possible samples of size 3 for this population.

(Use sampling without replacement).

b) Using the ten values, compute the mean and variance of .

c) Compute the mean and variance of the population. Compare your results to those in part b. Interpret your findings.

7. For a population, , and , find the Z values for each of the following for n =49.

a) ; b) ; c)

8. A population has a mean of 58 and a standard deviation of 12. Assuming

, find the following probabilities for a sample size of 50.

a) ; b)

9. Let X be a continuous random variable that has a distribution with and . Assuming , find the probability that the sample mean for a random sample of 64 taken from this population will be

a) less than 82.3;

b) more than 86.7.

10. The amount of electric bills for all households in a city have a skewed probability distribution with a mean of $65 and a standard deviation of $25. Find the probability that the sample mean amount of electric bills for a random sample of 75 households selected from this city will be

a) more than $70;

b) between $58 and $63;

c) within $6 of the population mean;

d) more than the population mean by at least $5?

11. The balances of all saving accounts at a local bank has a mean equal $45 360 and standard deviation equal to $5 900. Find the probability that the mean of a sample of 80 saving accounts selected from this bank will be

a) less than $46 500;

b) between $40 000 and $49 200;

c) within $1 800 of the population mean;

d) lower than the population mean by $1 200 or more.

12. The mean and standard deviation of the population are 55 and 7, respectively. Sample of 40 observations is selected randomly from this population.

a) What is the probability that sample mean will be between 54 and 56?

b) Find the shortest interval centered at 55, where will lie with

probability 0.95.

13. Consider a random sample of size n =100 from a population that has a standard deviation of .

a) Find the probability that the sample mean will lie within 2 units of the population mean-that is, .

b) Find the number k so that .

c) What is the probability that will differ from by more than 4 units?

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