# Mathematics

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This has 100 total points possible

1. **Probability (15 pts.)**

The following contingency table show the results of a nationwide poll of adults asked if they favor gun control, opposed gun control, or have no opinion on gun control.

**Opinion of Gun Control**

Gender | Favor | Oppose | No Opinion | Total |

Females | 88 | 96 | 16 | 200 |

Males | 64 | 96 | 40 | 200 |

Total | 152 | 192 | 56 | 400 |

Show your work for each of the following questions.

a. If a person is chosen at random from the above survey, what is the probability that he or she has no opinion about gun control? (3 pts.)

b. Given that a person surveyed is opposed to gun control, what is the probability that this person is male? (3 pts.)

c. Given that a person surveyed is female, what is the probability that she favors gun control? (3 pts.)

d. Is opposition to gun control independent of gender? (3 pts.)

e. Is favoring gun control independent of gender? Offer an explanation along with your calculations. (3 pts.)

**2. Binomial Distribution (15 pts)**

Assume that a fair, two-sided coin is tossed 3 times. The random variable X takes the value “1” when the outcome is a Head and takes the value “0” when it is a Tail. The random variable X follows a Binomial Distribution.

a. Graph the distribution of the random variable X below (provide the appropriate values for the random variable and probability associated with each value). (3 pts.)

P(X)

X

b. Find the Mean of this distribution (2 pts.)

c. Find the Variance of this distribution (2 pts.)

d. Find P(X = 2) (2 pts.)

e. Find P(X ≥ 1)

(2 pts.)

f. Find P(X > 1)

(2 pts.)

g. Find P(X > 1 | 1st toss = Head)

(2 pts.)

**3. Normal Distribution (12 pts.)**

During 2001, 61.3% of U.S. households purchased ground coffee and spent an average of $36.16 on ground coffee during the year. Consider the annual ground coffee expenditures for households purchasing ground coffee, assuming that these expenditures are approximately distributed as a normal random variable with a mean of $36.16 and a standard deviation of $10.

a. Find the probability that a household spent less than $25.(3pts.)

b. Find the probability that a household spent more than $50. (3pts.)

c. What proportion of the households spent between $30 and $40? (3pts.)

d. 99% of the households spent less than what amount? (3pts.)

**4. Hypotheses Testing: Two-Tailed Tests (20 pts.)**

You are a manager of a fast-food restaurant. You want to determine whether the population mean waiting time to place an order has changed in the past month from its previous population mean of 5.5 minutes. From past experience, you can assume that the population is normally distributed. You select a sample of 30 orders during a one-hour period. The sample mean is 5.2 minutes and the sample standard deviation is 1.4 minutes.

a. State the null and alternative hypotheses.(4 pts.)

b. At the 5% significance level can you reject the null hypothesis that the mean is 5.5. minutes? (show critical values and test statistic). (4 pts.)

c. Assume now that the sample size increases to 50. At the 5% significance level can you reject the null hypothesis that the mean is 5.5. minutes? (show critical values and test statistic). (4 pts.)

d. Assume now that the sample size is again at 30, but the level of significance is at 1%. Can you reject the null hypothesis that the mean is 5.5. minutes? (show critical values and test statistic). (4 pts.)

e. Assume that your sample size is 30, the level of significance is 5%, but now you find that you made a mistake in calculating sample standard deviation. The new sample standard deviation is 1.5 minutes. Can you reject the null hypothesis that the mean is 5.5. minutes? (show critical values and test statistic). (4 pts.)

**5. Simple Linear Regression (22 pts.)**

The following data is for square footage (X) and annual sales for a particular type of retail store in 4 different locations.

Store 1000 ft.2 (X) Annual Sales in millions $ (Y)

——- ————– ————————————

#1 3 7

#2 7 9

#3 1 5

#4 5 7

We can fit a linear regression model with the above data that follows the form

Ŷi = b0 + b1 Xi

a. Compute/Estimate b0 and b1 using the Least-Squares Method. (5 pts.)

b. Compute the Coefficient of Determination (r-squared). (4 pts.)

c. Given that the Standard Error of the Estimate SXY = 0.63246

(Note: this is not the SE of the coefficient)

(1) Is the slope coefficient b1 significant at the 10% level or better? (State the null and alternative hypotheses, state the appropriate critical t value and compute the t statistic). (5 pts.)

(2) Give an interpretation of b1. (use the numerical value of b1 in your explanation). (4 pts.)

(3) What are the degrees of freedom associated with the t-statistic used to test the null hypothesis for any one of the parameters? (4 pts.)

**6. Multiple Regression (16 pts.)**

A production function for the U.S. data can be expressed as Yt = AKt1Lt2

where A = Total factor productivity

Yt = Real output (in constant 1900 dollars)

Kt = Quantity of capital (in constant 1900 dollars)

Lt = Labor hours/week

This expression can be linearized by taking the natural logarithms of each side of this equation. Based on annual data for the U.S. manufacturing sector 1945-1994, a study estimates a production function using the logarithms of the data and obtains the following regression results,

**Dependent Variable: lnYt**

**Coefficients Standard Error**

**Intercept (b0) **2.81 1.38

**lnKt (b1) **0.53 0.27

**lnLt (b2) **0.91 0.14

**df SS F p-value**

**Regression **2 203693.3 6.745406 0.010884

Error 47 181184.1

**Total** _ _ _ _ _ _ _

1. What is the value of the SST? (3 pts.)

2. What are the degrees of freedom associated with SST? (3 pts.)

3. What is the r-squared? (3 pts.)

4. Is b1 significant at the 10% level? (show your work). (3 pts.)

5. Give an interpretation of b2 (Note that the variables are expressed as logarithms of the data). (4 pts.)