# Midterm1 Sol

**Topics:**Normal distribution, Variance, Regression analysis

**Pages:**12 (1304 words)

**Published:**December 1, 2014

March 5 (Tuesday), 2013

• Name:

• ID number:

• This exam must be your own work entirely. You can not talk to or share information with anybody. You are not allowed to share materials, and calculators. • Cell phone must be turned off.

• You have 50 minutes to complete the exam.

1

Problem 1 (21 points total, 3 points each)

Choose one of the listed choices for each question (no explanation is needed), put your answers in the table on page 4.

1. Suppose you have a sample x1 , x2 , . . . , xn from a population. Which of the following has different unit as sample mean? (A). population mean

(B). sample variance

(C). standard error

(D). single observation xi for i = 1, 2, . . . , n

2. In paired t-test with sample size n1 = n2 = 20, you have H0 : µ = 0 vs. Ha : µ < 0, the t-statistic is 3.4. what is the p-value?

(A).

1 - pt(3.4, 20)

(B).

1 - pt(3.4, 19)

(C).

pt(3.4, 20)

(D).

pt(3.4, 19)

3. Suppose y is response, x1 is numerical predictor, and x2 is categorical predictor with 2 levels. Which of the following R code will generate parallel line models? (A).

lm(y ~ 1)

(B).

lm(y ~ x1)

(C).

lm(y ~ x1 + x2)

(D).

lm(y ~ x1 * x2)

4. In paired t-test with H0 : µ = 2, a 95% confidence interval for the mean of difference is (-0.035, 0.057). The corresponding t-test (two-sided Ha ) would: (A). reject H0 at the α=0.05 significance level.

2

(B). fail to reject H0 at the α=0.05 significance level.

(C). can’t tell without more information.

5. For what type of experiment you can make causal inference, according to chapter 1 in the textbook?

(A). Randomized experiment

(B). Observational experiment

(C). Neither

(D). Both

6. In simple linear regression yi = β0 + β1 xi + i , where i = 1, 2, . . . , n. Define residual ei = yi − yˆi , where yˆi are fitted values. Which of the following is correct? (A). E( i ) = 0

(B).

n

i=1 i

=0

(C).

n

i=1 ei

=0

(D). (A), (B), (C) are all wrong

7. In the following table, which cell corresponds to the type I Error?

Reject H0

Do not reject H0

H0 is true

(1)

(3)

(A). (1)

(B). (2)

(C). (3)

(D). (4)

3

Ha is true

(2)

(4)

Put your answers for multiple choice problems here (use capital letters):

1

B

2 3

D C

4 5

A A

6

B

7

A

Problem 2 (36 points, 4 points each)

Based on the following output of linear regression, answer questions 1 - 2:

> summary(m)

Call:

lm(formula = taste ~ lactic + h2s, data = data)

Residuals:

Min

1Q

-17.343 -6.530

Median

-1.164

3Q

4.844

Max

25.618

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -27.592

8.982 -3.072 0.00481 **

lactic

19.887

7.959

2.499 0.01885 *

h2s

3.946

1.136

(1) 0.00174 **

--Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1

1

Residual standard error: 9.942 on 27 degrees of freedom

Multiple R-squared: 0.6517,Adjusted R-squared: 0.6259

F-statistic: 25.26 on 2 and 27 DF, p-value: 6.551e-07

4

1. Calculate (1) in the output?

ˆ

3.946

Answer: SE(β2βˆ ) = 1.136

= 3.47

2

2. What is the proportion of variation in taste that is NOT explained by the variables h2s and lactic?

Answer: 1 − 0.6517 = 0.3483. So 34.83% of variation is explained.

3. We have two random variables x and y, then under what condition we have var(x − y) = var(x) + var(y), where var stands for variance? Answer: x and y are independent.

4. What are the four model assumptions in linear regression? Answer: 1) Linearity, 2) Normality, 3) Constant variance, 4) Independence

5. Under what conditions, you will prefer non-parametric test over standard t-test? Answer: Samples don’t come from a distribution; Data are heavily skewed; Some outliers are found.

x

¯−µ

6. If x1 , x2 , . . . , xn follow N (µ, sd = σ), then what distribution does SE(¯ follow?

x)

x

¯−µ

And what distribution does SD(¯x) follow?

Answer: 1) t-distribution with d.f. = n − 1; 2) standard normal distribution.

7....

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