Deck 12: Linear Regression and Correlation

Question 1

If there is a negative correlation between the independent variable x and the dependent variable y, then to test this, the appropriate null and alternative hypotheses would be .

Answer 1

False

Question 2

In regression analysis, a careful study of the differences, , between observed and estimated y values, given x (in order to decide whether crucial assumptions are fulfilled that allow valid inferences about the true regression line to be made from an estimated regression line) is called residual analysis.

Answer 2

True

Question 3

Large coefficient of determination value will result in a small standard error of the estimate for the regression model, thus providing prediction intervals that are narrow.

Answer 3

False

Question 4

In developing a 90% confidence interval for the average value of y from a simple linear regression problem involving 12 observations, the appropriate table value would be 1.796.

Question 5

If the correlation coefficient for two variables is found to be .094, then the scatterplot will show the data upward sloping from lower left to upper right.

Question 6

The regression model = 36.5 + 20.1x has been computed based on a sample of 50 observations. One observation in the sample was (x, y) = (14, 350.9). Given this, the residual value for this observation is 33.

Question 7

In simple linear regression, one can use the plot of residuals versus the fitted values of y to check for a constant variance as well as to make sure that the linear model is in fact adequate.

Question 8

If the correlation coefficient between two variables is very close to zero, this means that there is no relationship between the two variables.

Question 9

Given that n = 37, and the value of sample Spearman rank correlation coefficient = 0.35, the value of the test statistic for testing .

Question 10

The normal probability plot is a graph that plots the residuals against the expected value of that residual if it had come from a normal distribution. When the residuals are normally distributed or approximately so, the plot should appear as a straight line, sloping upward at a 45° angle.

Question 11

Simple regression analysis is a statistical technique that establishes an index that provides, in a single number, a measure of the strength of association between two variables.

Question 12

In simple linear regression analysis, if the correlation coefficient between the independent variable x and the dependent variable y is -.85, this means that the scatterplot generated by the same data values would show points that would fall on a straight line with slope equal to -.85.

Question 13

A perfect correlation between two variables will always produce a correlation coefficient of + 1.0.

Question 14

In a simple linear regression problem, the least squares line is = 2.73 - 1.02x, and the coefficient of determination is 0.7744. The correlation coefficient must be -0.88.

Question 15

The prediction interval developed from a simple linear regression model will be very narrow when the value of x used to predict y is equal to the mean value .

Question 16

When regression analysis is used for prediction, the confidence interval for the average y given x will be wider than the prediction interval for a particular value of y given x.

Question 17

In regression analysis, a graph of each residual against the corresponding fitted value is called a scatter diagram.

Question 18

In a simple linear regression analysis, it was stated that the correlation between starting salary and years of experience is 0.80. This indicates that 80% of the variation in starting salary is explained by years of experience.

Question 19

If all the points in a scatterplot lie on the least squares regression line, then the correlation coefficient must be 1.0

Question 20

In simple linear regression analysis, if the independent variable x and the dependent variable y are highly correlated, this does not only mean that they are linearly related, but it also means that a change in x will cause a change in y.

Question 21

In the simple linear regression model which of the following is true regarding the values of the random error term ?

A) They are independent.
B) They have a mean of 0 and a common variance, independent of x.
C) They are normally distributed.
D) All of these.
E) None of these.

Question 22

Which of the following statements about simple correlation analysis are correct?

A) When all the points in a scatter diagram lie precisely on the estimated regression line, the sample coefficient of correlation will equal 0.
B) When all the points in a scatter diagram lie precisely on the estimated regression line, the sample coefficient of correlation will show the variables to be perfectly correlated.
C) When all the points in a scatter diagram are so widely scattered as to make x completely worthless as a predictor of y, the sample coefficient of correlation will equal 1.
D) All of these.
E) None of these.

Question 23

The confidence interval estimate of the expected value of y will be wider than the prediction interval for the same given value of x and confidence level. This is because there is more error in estimating a mean value as opposed to predicting an individual value.

Question 24

The confidence interval estimate of the expected value of y will be narrower than the prediction interval for the same given value of x and confidence level. This is because there is less error in estimating a mean value as opposed to predicting an individual value.

Question 25

In developing a 80% prediction interval for the particular value of y from a simple linear regression problem involving a sample of size 12, the appropriate table value would be 1.372.

Question 26

The following are coefficients of correlation (r). The one that indicates a strong positive linear relationship between the two variables of interest is:

A) 0.8
B) -0.9
C) 0.9
D) -1.3
E) -1

Question 27

Given the least squares regression line = -4.63 + 1.38x, and a coefficient of determination of 0.9025, the correlation coefficient must be:

A) -0.95
B) +0.95
C) +1.38
D) -0.81
E) 0.81

Question 28

An indication of no linear relationship between two variables x and y would be:

A) a coefficient of correlation of 1
B) a coefficient of correlation of 0
C) a coefficient of correlation of -1
D) a coefficient of determination of 1
E) 0.5

Question 29

In the simple linear regression model which of the following is false regarding the values of the random error term ?

A) They are independent.
B) They have a mean of 0 and a variance of 1, independent of x.
C) They are normally distributed.
D) None of these.
E) All of these.

Question 30

Which of the following statements is false regarding the residuals in simple linear regression model?

A) They sum to 0.
B) They have a mean of 0.
C) They have a standard deviation of 1.
D) None of these.
E) All of these.

Question 31

In developing a 90% confidence interval for the expected value of y from a simple linear regression problem involving a sample of size 15, the appropriate table value would be 1.761.

Question 32

In developing 90% prediction interval for the particular value of y from a simple linear regression problem involving a sample of size 14, the appropriate table value would be 2.179.

Question 33

In simple linear regression, the plot of residuals versus fitted values 3 can be used to check for:

A) normality
B) a constant variance independent of x
C) independence
D) all of these
E) none of these

Question 34

In a simple linear regression problem including n = 10 observations, which of the following table values would be appropriate for a 95% confidence interval estimation for the average value of y?

A) 2.228
B) 2.262
C) 2.306
D) 1.860
E) 18.60

Question 35

In a regression problem the following pairs of (x, y) are given: (4, 1), (4, -1), (4, 0), (4, -2) and (4, 2). That indicates that:

A) the correlation coefficient is -1
B) the correlation coefficient is 0
C) the correlation coefficient is 1
D) the coefficient of determination is between -2 and 2
E) none of these

Question 36

The following are coefficients of correlation (r). The one that indicates a strong negative linear relationship between the two variables of interest is:

A) 0.8
B) -0.9
C) 0.9
D) -1.3
E) 1

Question 37

In developing a 95% confidence interval for the expected value of y from a simple linear regression problem involving a sample of size 10, the appropriate table value would be 1.86.

Question 38

In order to predict with 95% confidence a particular value of y for a given value of x in a simple linear regression problem, a random sample of 20 observations is taken. The appropriate table value that would be used is 2.101.

Question 39

In publishing the results of some research work, the following values of the correlation coefficient were listed. Which one would appear to be incorrect?

A) 0.95
B) 0.05
C) 0.00
D) 1.05
E) 0.11

Question 40

In simple linear regression, the plot of residuals versus fitted values should:

A) be free of any patterns
B) appear as a random scatter of points about 0 on the vertical axis
C) approximately have the same vertical spread for all values
D) all of these
E) none of these

Question 41

A university admissions committee was interested in examining the relationship between a student's score on the SAT exam, x, and the student's grade point average, y, (GPA) at the end the student's freshman year of college. The committee selected a random sample of 25 students and recorded the SAT score and GPA at the end of the freshman year of college for each student. Use the following output that was generated using Minitab to answer the questions below: Determine the correlation between a student's SAT score and GPA at the end of the freshman year.
Since b is ______________ the correlation is ______________.
Interpret the value.
There is a ______________ linear relationship between a student's SAT score and GPA at the end of the freshman year.

Question 42

A microwave manufacturer has collected the data shown below on number of units sold (y) in the thousands of dollars and the number of ads (x) placed during the month. Calculate the preliminary sums of squares and cross-products.
S_xx = ______________
S_yy = ______________
S_xy = ______________
Calculate:
SSE = ______________
MSE = ______________
Determine the least-squares regression line. = ______________
Compute a point estimate of number of units sold if there are 140 ads.
______________
Compute the standard error of the point estimate of number of units sold if there are 140 ads.
______________
Compute a 95% confidence interval for the average number of units sold in all months with 140 ads.
______________ Enter (n1, n2)
Compute a 95% prediction interval for sales during the next month that happens to be associated with 140 ads.
______________ Enter (n1, n2)

Question 43

Six points have these coordinates: The normal probability plot and the residuals versus fitted values plots generated by Minitab are shown below. Does it appear that any regression assumptions have been violated?
______________
Explain.
________________________________________________________

Question 44

The manager of an ice cream store is interested in examining the relationship between sales of ice cream (in gallons per day) and maximum temperature of the day. The vendor records the following data for a random sample of five days in the summer, where y is number of gallons of ice cream sold per day and x is maximum temperature, in degrees Fahrenheit, recorded for the day: The following summary information were computed: Find and interpret the correlation between maximum daily temperature and daily sales of ice cream.
S_yy = ______________
What is the correlation coefficient?
r = ______________
There is ______________ linear relationship between daily sales of ice cream and maximum daily temperature.

Question 45

If a sample of 25 observations is selected, and the sample correlation coefficient between the variables x and y is r = 0.525, what is the test statistic value for testing

A) About 3.65.
B) About 2.96.
C) About 3.08.
D) About 3.81.
E) About 3.96.

Question 46

In order to predict with 98% confidence the expected value of y for a given value of x in a simple linear regression problem, a random sample of 15 observations is taken. Which of the following t-table values listed below would be used?

A) 1.350
B) 1.771
C) 2.160
D) 2.650
E) 1.750

Question 47

In order to estimate with 95% confidence the expected value of y for a given value of x in a simple linear regression problem, a random sample of 10 observations is taken. Which of the following t-table values listed below would be used?

A) 2.228
B) 2.306
C) 1.860
D) 1.812
E) 2.812

Question 48

If the true correlation between two variables is zero, then which of the following statements is true?

A) There is no linear relationship between the two variables.
B) There may be no relationship between the two variables.
C) Neither "There is no linear relationship between the two variables" nor "There may be no relationship between the two variables".
D) Both "There is no linear relationship between the two variables" and "There may be no relationship between the two variables".

Question 49

Given a specific value of x and confidence level, which of the following statements is correct?

A) The confidence interval estimate of the expected value of y can be calculated but the prediction interval of y for the given value of x cannot be calculated.
B) The confidence interval estimate of the expected value of y will be wider than the prediction interval.
C) The prediction interval of y for the given value of x can be calculated but the confidence interval estimate of the expected value of y cannot be calculated.
D) The confidence interval estimate of the expected value of y will be narrower than the prediction interval.
E) None of these.

Question 50

In order to predict with 90% confidence the expected value of y for a given value of x in a simple linear regression problem, a random sample of 10 observations is taken. Which of the following t-table values listed below would be used?

A) 2.228
B) 2.306
C) 1.860
D) 1.812
E) 2.812

Question 51

The confidence interval estimate of the expected value of y for a given value y x, compared to the prediction interval of y for the same given value of x and confidence level, will be:

A) wider
B) narrower
C) the same
D) impossible to know

Question 52

In regression analysis we use the Spearman rank correlation coefficient to measure and test to determine whether a relationship exists between the two variables if:

A) one or both variables may be ordinal
B) both variables are interval but the normality requirement is not met
C) both one or both variables may be ordinal and both variables are interval but the normality requirement is not met
D) neither one or both variables may be ordinal nor both variables are interval but the normality requirement is not met
E) none of these

Question 53

In order to predict with 99% confidence the expected value of y for a given value of x in a simple linear regression problem, a random sample of 10 observations is taken. Which of the following t-table values listed below would be used?

A) 1.860
B) 2.306
C) 2.896
D) 3.355
E) 2.355

Question 54

The width of the confidence interval estimate for the predicted value of y depends on:

A) the standard error of the estimate
B) the value of x for which the prediction is being made
C) the sample size
D) all of these
E) none of these

Question 55

A company manager is interested in the relationship between x = number of years that an employee has been with the company and y = the employee's annual salary (in thousands of dollars). The following statistical software output is from a regression analysis for predicting y from x for n = 15 data points. Find the correlation coefficient.
r = ______________
There is ______________ linear relationship between x and y.

Question 56

A study of 20 students showed that the correlation between the time spent writing a test and the number of hours studied the night before the test was 0.35. Using a level of significance equal to 0.05, which of the following statements is true?

A) The sample correlation coefficient could be zero since the test statistic does not fall in the rejection region.
B) The null hypothesis that the population mean is equal to zero should not be rejected, and we should conclude that the true correlation coefficient is zero.
C) There is not enough statistical evidence to conclude that the true correlation coefficient is different from zero.
D) The null hypothesis that the population variance is equal to zero should be rejected, and we should conclude that the true correlation coefficient is zero.
E) None of these.

Question 57

In studying the relationship between two variables x and y, a scatterplot can be used to detect which of the following?

A) A positive linear relationship.
B) A negative linear relationship.
C) A relationship that is not linear.
D) All of these.
E) None of these.

Question 58

If the plot of the residuals is fan shaped, which assumption of regression analysis if violated?

A) Normality.
B) Homoscedasticity.
C) Independence of errors.
D) No assumptions are violated, the graph should resemble a fan.
E) All of these.

Question 59

The general manager of a chain of furniture stores believes that experience is the most important factor in determining the level of success of a salesperson. To examine this belief she records last month's sales (in $1,000s) and the years of experience of 10 randomly selected salespeople. These data are listed below. Predict with 95% confidence the monthly sales of a salesperson with 10 years of experience.
CI = ______________ Enter (n1, n2) in thousands
Estimate with 95% confidence the average monthly sales of all salespersons with 10 years of experience.
CI = ______________ Enter (n1, n2) in thousands
Which interval in the previous two questions is narrower: the confidence interval estimate of the expected value of y or the prediction interval for the same given value of x (10 years) and same confidence level?
______________
Why?
________________________________________________________

Question 60

In order to predict with 80% confidence the expected value of y for a given value of x in a simple linear regression problem, a random sample of 15 observations is taken. Which of the following t-table values listed below would be used?

A) 1.350
B) 1.771
C) 2.160
D) 2.650
E) 2.260

Question 61

One way to measure the strength of the relationship between the response variable y and the predictor variable x is to calculate the coefficient of determination; that is, the proportion of the total variation in y that is explained by the linear regression of y on x.

Question 62

Regression analysis is a statistical method that seeks to establish an equation that allows the unknown value of one variable to be estimated from the known value of one or more other variables.

Question 63

An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief she gathers data about the last eight winners of her favorite game show. She records their winnings in dollars and the number of years of education. The results are as follows. Predict with 95% confidence the winnings of a contestant who has 15 years of education.
CI = ______________ Enter (n1, n2)
Predict with 95% confidence the winnings of a contestant who has 10 years of education.
CI = ______________ Enter (n1, n2)
Estimate with 95% confidence the average winnings of all contestants who have 15 years of education.
CI = ______________ Enter (n1, n2)
Estimate with 95% confidence the average winnings of all contestants who have 10 years of education.
CI = ______________ Enter (n1, n2)

Question 64

The vertical spread of the data points about the regression line is measured by the y-intercept.

Question 65

In regression analysis, the independent variable is a variable whose value is known and is being used to explain or predict the value of another variable.

Question 66

A regression analysis between sales (in $1000) and advertising (in $100) resulted in the following least squares line: = 77 +8x. This implies that if advertising is $600, then the predicted amount of sales (in dollars) is $125,000.

Question 67

In simple linear regression, if the estimated values and the corresponding actual values are equal, then the standard error of estimate, SE( ), must equal -1.0.

Question 68

If a least squares regression line has a y-intercept of 6.84 and a slope of 2.16, then when x = 1 the actual value of y must be 9.

Question 69

The value of the sum of squares for regression (SSR) can never be larger than 100.

Question 70

A professor of economics wants to study the relationship between income (y in $1000s) and education (x in years). A random sample eight individuals is taken and the results are shown below. Predict with 95% confidence the income of an individual with 10 years of education.
CI = ______________ Enter (n1, n2) in thousands
Estimate with 95% confidence the average income of all individuals with 10 years of education.
CI = ______________ Enter (n1, n2) in thousands
Which interval in the previous two questions is narrower: the confidence interval estimate of the expected value of y or the prediction interval for the same given value of x (10 years) and same confidence level?
______________
Why?
________________________________________________________

Question 71

A regression analysis between weight (y in pounds) and height (x in inches) resulted in the following least squares line: = 135 + 6x. This implies that if the height is increased by 1 inch, the weight is expected to increase by an average of 6 pounds.

Question 72

A regression analysis between sales (in $1000) and advertising (in $) resulted in the following least squares line: = 60 + 5x. This implies that an increase of $1 in advertising is expected to result in an increase of $65 in sales.

Question 73

The residuals are observations of the error variable . Consequently, the minimized sum of squared deviations is called the sum of squares for error, denoted SSE.

Question 74

The value of the sum of squares for error (SSE) can never be larger than the total sum of squares (Total SS).

Question 75

Given that the sum of squares for error (SSE) is 52 and the sum of squares for regression (SSR) is 148, then the coefficient of determination is 0.74.

Question 76

In a simple linear regression setting, the probabilistic model equation allows for some deviation of the points about the regression line, making it a more practical model.

Question 77

The sum of squares for regression (SSR) can never be larger than the sum of squares for error (SSE).

Question 78

If the coefficient of determination is 0.982, then the slope of the regression line must be positive.

Question 79

The method of least squares requires that the sum of the squared deviations between actual y values in the scatter diagram and y values predicted by the regression line be minimized.

Question 80

In a simple linear regression setting, the deterministic model equation determines an exact value of the dependent variable y when the value of the independent variable x is given, since all points must lie exactly on the line.