Multiple Choice Exercise

*** Choose one answer and click [Submit] button

12.1 The variables X and Y have a strong relationship with a quadratic equation () as shown in the following table. What is their sample correlation coefficient?

XY
... ...
-3 9
-2 4
-1 1
0 0
1 1
2 4
3 9
... ...





        

12.2 Which is a wrong description of the sample correlation coefficient \(r\)?





        

12.3 Which is a right description of the sample correlation coefficient \(r\)?





        

12.4 If the sample correlation coefficient between \(x\) and \(y\) is \(r\), what is the sample correlation coefficient between \(2x\) and \(3y +1\)?





        

12.5 Find the sample correlation coefficient between \(x\) and \(y\) of the following data.

\(x\)\(y\)
10 2
20 4
30 6
40 8





        

12.6 If the correlation coefficient of two variables \(x, y\) is 0, what is the right description?





        

12.7 Which one of the following descriptions on the sample correlation coefficient \(r\) is not right?





        

12.8 Find the sample correlation coefficient between \(x\) and \(y\) of the following data.

\(x\)\(y\)
1 5
2 4
3 3
4 2
5 1





        

12.9 If \(X\) and \(Y\) are independent, what is the sample correlation coefficient \(r\)?





        

12.10 Which one of the followings is not right for description of the sample correlation coefficient \(r\) between \(X\) and \(Y\)?





        

12.11 Which one of the followings has positive correlation?





        

12.12 Find the sample covariance between \(x\) and \(y\) of the following data.

\(x\)\(y\)
1 5
2 5
3 5
4 5





        

12.13 Find the sample covariance between \(x\) and \(y\) of the following data.

\(x\)\(y\)
1 6
2 8
3 10
4 12
5 14





        

12.14 If the standard deviations of the \(X\) and \(Y\) variables are 4.06 and 2.65 respectively, the covariance is 10.50, what is the sample correlation coefficient \(r\)?





        

12.15 Find the regression line between \(x\) and \(y\) using the following data.

\(x\)\(y\)
1 1
2 4
3 7
4 10
5 13





        

12.16 If we know the sample correlation coefficient \(r\) and the standard deviations of \(X\) and \(Y\), \(s_x\) and \(s_y\) respectively, what is the regression line equation?





        

12.17 If the sample correlation coefficient of two random variables \(x\) and \(y\) is \(r\) = 0.5, the sample means are \(\small \overline x = 10, \overline y = 14\), and the sample standard deviations are \(s_x = 2, s_y = 3\), what is the regression line of \(y\) on \(x\)?





        

12.18 Find the regression coefficient of the regression line using the following data.

sample meansample standard deviationcorrelation coefficient
\(X\) 40 4 0.75
\(Y\) 30 3





        

12.19 Which one of the following statements is true about the regression line of two variables \(\small X\) and \(\small Y\), the regression line of \(\small Y\) on \(\small X\) and the regression line of \(\small X\) on \(\small Y\)?





        

12.20 Find the regression coefficient \(b\) of the regression line \(\small Y = a + bX\) using the following data.

sample meansample standard deviationcorrelation coefficient
\(\small X\) 12 3 0.6
\(\small Y\) 13 4





        

12.21 Which one is a wrong explanation about the regression coefficient \(b\) and the sample correlation coefficient \(r\)?





        

12.22 If a regression line is \(\small Y = 4 + 0.4X\) and the sample standard deviations of \(\small X\) and \(\small Y\) are 4, 2 respectively, what is the value of the sample correlation coefficient \(r\)?





        

Practice Exercise

*** Answer the followings

12.1 A survey was conducted on the level of education(\(\small X\), the period after graduating a high school, unit: year) for 10 businessmen and annual income (\(\small Y\), unit: one thousand USD) after graduating from the high school.

id Education period
(\(\small X\))		 Annual income
(\(\small Y\))
1 4 50
2 2 37
3 0 35
4 3 45
5 4 57
6 4 49
7 5 60
8 5 47
9 2 39
10 2 50

1) Draw a scatter plot of data and interpret.
2) Calculate the sample correlation coefficient.
3) Apply the regression analysis with annual income as the dependent variable and the level of education as the independent variable.

12.2 The following data shows studying time for a week (\(\small X\)) and the grade (\(\small Y\)) of six students.

Studying time (\(\small X\)) Grade (\(\small Y\))
15 2.0
28 2.7
13 1.3
20 1.9
4 0.9
10 1.7

1) Find a regression line and 95% confidence interval for (it is a further grade score that is expected to be raised when a student studies one more hour a week.)
2) Calculate a 99% confidence interval in the average score of a student who studies an average of 12 hours a week.
3) Test for hypothesis \(\small H_0 : \beta = 0.10 , H_1 : \beta < 0.10\), (significance level α = 0.01).

12.3 A professor of statistics argues that a student’s final test score can be predicted from his/her midterm. Five students were randomly selected and their mid-term and final exam scores are as follows:

id Midterm (\(\small X\)) Final (\(\small Y\))
1 92 87
2 65 71
3 75 75
4 83 84
5 95 93

1) Draw a scatter plot of this data with mid-term score on X axis and final score on Y axis. What do you think is the relationship between mid-term and final scores?
2) Find the regression line and analyse the result.

12.4 An economist argues that there is a clear relationship between coffee and sugar prices. 'When people buy coffee, they will also buy sugar. Isn't it natural that the higher the demand, the higher the price?' We collected the following sample data to test his theory.

Year Coffee price Sugar Price
1985 0.68 0.245
1986 1.21 0.126
1987 1.92 0.092
1988 1.81 0.086
1989 1.55 0.101
1990 1.87 0.223
1991 1.56 0.212

1) Prepare a scatter plot with the coffee price on \(\small X\) axis and sugar price on \(\small Y\) axis. Is this data true to this economist's theory?
2) Test this economist's theory by using a regression analysis.

12.5 A rope manufacturer thinks that the strength of the rope is proportional to the nylon content of the rope. Ten ropes are randomly selected and their data are as follows:

% Nylon (\(\small X\)) AStrength (psi) (\(\small Y\))
0 260
10 360
20 490
20 510
30 600
30 600
40 680
50 820
60 910
70 990

1) Draw a scatter plot with the % Nylon on X axis and strength on Y axis. Find a regression line using the least squares method. Draw this estimated regression line on the scatter plot.
2) Estimate the strength of a rope in case of 33% nylon.
3) Estimate the strength of a rope in case of 66% nylon.
4) The strength of two ropes in case of 20% nylon on the data are different. How can you explain this variation in a regression model?
5) Estimate the strength of a rope in case of 0% nylon. Why is this estimate different from the observed value of 260?
6) Obtain a 95% confidence interval for the strength of the 0% nylon rope.
7) If the observed strength of the 0% nylon rope was outside the confidence interval in 6), how would you interpret this result?

12.6 A health scientist randomly selected 20 people to determine the effects of smoking and obesity on their physical strength and examined the average daily smoking rate (\(x_1\), number/day), the ratio of weight by height (\(x_2\), kg/m), and the time to continue to exercise with a certain intensity (\(y\), in hours). Test whether smoking and obesity can affect your exercising time with a certain intensity. Apply a multiple regression model by using 『eStat』.

smoking rate
\(x_1\)		 ratio of weight by height
\(x_2\)		 Atime to continue to exercise
\(y\)
24 53 11
0 47 22
25 50 7
0 52 26
5 40 22
18 44 15
20 46 9
0 45 23
15 56 15
6 40 24
0 45 27
15 47 14
18 41 13
5 38 21
10 51 20
0 43 24
12 38 15
0 36 24
15 43 12
12 45 16

12.7 The price of old watches in an antique auction is said to be determined by the year of making the watch and the number of bidders. In order to see if this is true, the 32 recently auctioned alarm clocks were examined for the elapsed period (in years) after manufacture, the number of bidders and the auction price (in 1,000USD) as follows: Test the hypothesis that the auction price of the alarm clock increases with the increase in the number of bidders using the multiple linear regression model. (significance level: 0.05)

Elapsed Period
\(x_1\)		 Number of bidders
\(x_2\)		 Auction Price
\(y\)
127 13 1235
115 12 1080
127 7 845
150 9 1522
156 6 1047
182 11 1979
156 12 1822
132 10 1253
137 9 1297
113 9 946
137 15 1713
117 11 1024
137 8 1147
153 6 1092
117 13 1152
126 10 1336
170 14 2131
182 8 1550
162 11 1884
184 10 2041
143 6 854
159 9 1483
108 14 1055
175 8 1545
108 6 729
179 9 1792
111 15 1175
187 8 1593
111 7 785
115 7 744
194 5 1356
168 7 1262