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Regression, testing of hypothesis

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1. An investor owns three common stocks. Each stock, independent of the others, has equally likely chances of (1) increasing in value, (2) decreasing in value, or (3) remaining the same value. List the possible outcomes of this experiment. Estimate the probability at least two of the stocks increase in value.

2. If you ask three strangers about their birthdays, what is the probability: (a) All were born on Wednesday? (b) All were born on different days of the week? (c) None were born on Saturday?

3. It is estimated that 10 percent of those taking the quantitative methods portion of CPA examination fail that section. Sixty students are taking the exam on this Saturday.
a. How many would you expect to fail? What is the standard deviation?
b. What is the probability that exactly two students will fail?
c. What is the probability at least two students will fail?

4. A population of unknown shape has a mean of 75. You select a sample of 40. The standard deviation of the sample is 5. Compute the probability the sample mean is:
a. Less than 74.
b. Between 74 and 76
c. Between 76 and 77.
d. Greater than 77.

5. Information from the American Institute of Insurance indicates the mean amount of life insurance per household in the United States is $110,000. This distribution is positively skewed. The standard deviation of the population is not known.
a. A random sample of 50 households revealed a mean of $112,000 and a standard deviation of $40,000. What is the standard error of the mean?
b. Suppose that you selected 50 samples of households. What is the expected shape of the distribution of the sample mean?
c. What is the likelihood of selecting a sample with a mean of at least $112,000?
d. What is the likelihood of selecting a sample with a mean of more than $100,000?
e. Find the likelihood of selecting a sample with a mean of more than $100,000 but less than $112,000.

6. Given the following hypothesis:
H0: μ = 100
H1: μ = 100
A random sample of six resulted in the following values: 118, 105, 112, 119, 105 and 111.
Using the .05 significance level, can we conclude the mean is different from 100?
a. State the decision rule.
b. Compute the value of the test statistic.
c. What is your decision regarding the null hypothesis?
d. Estimate the p-value.

7. The amount of money out of income spent on housing in an important component of the cost of living. The total costs of housing for homeowners might include mortgage payments, property taxes, and utility costs (water, heat, electricity). An economist selected a sample of 20 homeowners in New England and then calculated these total housing costs as a percent of monthly income, five years ago and now. The information is reported below. Is it reasonable to conclude the percent is less now than five years ago?

Homeowner Five Years Ago Now Homeowner Five Years Ago Now
1 17 10 11 35 32
2 20 39 12 16 32
3 29 37 13 23 21
4 43 27 14 33 12
5 36 12 15 44 40
6 43 41 16 44 42
7 45 24 17 28 22
8 19 26 18 29 19
9 49 28 19 39 35
10 49 26 20 22 12