Question

unit 2 test

1. a local company is interested in supporting environmentally friendly initiatives such as carpooling among employees. the company surveyed all of the 200 employees at the downtown offices. employees responded as to whether or not they own a car and to the location of the home where they live. the results are shown in the table below.

		location of home
car ownership	yes	10	15	35	60
	no	60	55	25	140
	total	70	70	60	200

which of the following statements about a randomly chosen person from these 200 employees is true?
(a) if the person owns a car, he or she is more likely to live elsewhere in the city than to live in the downtown area in the city.
(b) if the person does not own a car, he or she is more likely to live outside the city than to live in the city (downtown area or elsewhere).
(c) the person is more likely to own a car if he or she lives in the city (downtown area or elsewhere) than if he or she lives outside the city.
(d) the person is more likely to live in the downtown area in the city than elsewhere in the city.
(e) the person is more likely to own a car than not to own a car.

1. consider n pairs of numbers (x₁,y₁), (x₂,y₂), ..., and (xₙ, yₙ). the mean and standard deviation of the x-values are \\(\bar{x} = 5\\) and \\(s_x = 4\\), respectively. the mean and standard deviation of the y-values are \\(\bar{y} = 10\\) and \\(s_y = 10\\) respectively. of the following, which could be the least squares regression line?

(a) \\(\hat{y} = -5.0 + 3.0x\\)
(b) \\(\hat{y} = 3.0x\\)
(c) \\(\hat{y} = 5.0 + 2.5x\\)
(d) \\(\hat{y} = 8.5 + 0.3x\\)
(e) \\(\hat{y} = 10.0 + 0.4x\\)

Explanation:

Question 1

Step1: Analyze Option A

For car owners (60 people), number living downtown: 10, elsewhere: 15. Since \( 15 > 10 \), more likely to live elsewhere.

Step2: Analyze Option B

Non - car owners (140 people): live in city (downtown + elsewhere) = \( 60 + 55=115 \), outside city: 25. \( 115>25 \), so less likely to live outside.

Step3: Analyze Option C

City dwellers (140 people): car owners = \( 10 + 15 = 25 \). Outside city (60 people): car owners = 35. \( 35>25 \), so less likely to own car in city.

Step4: Analyze Option D

Downtown population: 70, elsewhere in city: 70. Equal likelihood.

Step5: Analyze Option E

Car owners: 60, non - car owners: 140. \( 60<140 \), less likely to own car.

Step1: Recall Least Squares Line Property

The least squares regression line \( \hat{y}=a + bx \) passes through \( (\bar{x},\bar{y}) \), i.e., when \( x = \bar{x}=5 \), \( \hat{y}=\bar{y}=10 \). Also, the slope \( b \) satisfies \( |b|\leq\frac{s_y}{s_x} \) (by the property of correlation and slope, \( |r|\leq1 \) and \( b = r\frac{s_y}{s_x} \)). Here, \( \frac{s_y}{s_x}=\frac{10}{4} = 2.5 \), so \( |b|\leq2.5 \).

Step2: Check Each Option

• Option A: When \( x = 5 \), \( \hat{y}=- 5+3\times5=10 \). Slope \( 3>2.5 \), invalid.
• Option B: When \( x = 5 \), \( \hat{y}=3\times5 = 15

eq10 \), invalid.

• Option C: When \( x = 5 \), \( \hat{y}=5 + 2.5\times5=5 + 12.5 = 17.5

eq10 \), invalid.

• Option D: When \( x = 5 \), \( \hat{y}=8.5+0.3\times5=8.5 + 1.5 = 10 \). Slope \( 0.3\leq2.5 \), valid.
• Option E: When \( x = 5 \), \( \hat{y}=10+0.4\times5=10 + 2 = 12

eq10 \), invalid.

Answer:

A

Question 2