Homework answers / question archive / 1) If we find that there is a linear correlation between the concentration of carbon dioxide in our atmosphere and the global temperature, does that indicate that changes in the concentration of carbon dioxide cause changes in the global temperature? Choose the correct answer below

1) If we find that there is a linear correlation between the concentration of carbon dioxide in our atmosphere and the global temperature, does that indicate that changes in the concentration of carbon dioxide cause changes in the global temperature? Choose the correct answer below

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1) If we find that there is a linear correlation between the concentration of carbon dioxide in our atmosphere and the global temperature, does that indicate that changes in the concentration of carbon dioxide cause changes in the global temperature?

Choose the correct answer below.

A. No. The presence of a linear correlation between two variables does not imply that one of the variables is the cause of the other variable.

B. Yes. The presence of a linear correlation between two variables implies that one of the variables is the cause of the other variable.

2. For a sample of eight bears, researchers measured the distances around the bears' chests and weighed the bears. Minitab was used to find that the value of the linear correlation coefficient is r=0.727. Using a = 0.05, determine if there is a linear correlation between chest size and weight. What proportion of the variation in weight can be explained by the linear relationship between weight and chest size?

Click here to view a table of critical values for the correlation coefficient. '

a. Is there a linear correlation between chest size and weight?

A. Yes, because the absolute value of the test statistic exceeds the critical value of 0.707.

B. No, because the absolute value of the test statistic exceeds the critical value of 0.707.

C. Yes, because the test statistic falls between the critical values of - 0.707 and 0.707.

D. The answer cannot be determined from the given information.

b. What proportion of the variation in weight can be explained by the linear relationship between weight and chest size?

__________________ (Round to three decimal places as needed.)

1: Table of Critical Values

n	a = 0.5	a = .01
4	.950	.990
5	.878	.959
6	.811	.917
7	.754	.875
8	.707	.834
9	.666	.798
10	.632	.765
11	.602	.735
12	.576	.708
13	.553	.684
14	.532	.661
15	.514	.641
16	.497	.623
17	.482	.606
18	.468	.590
19	.456	.575
20	.444	.561
25	.396	.505
30	.361	.463
35	.335	.430
40	.312	.402
45	.294	.378
50	.279	.361
60	.254	.330
70	.236	.305
80	.220	.286
90	.207	.269
100	.196	.256

 

NOTE: To test H₀: r = O against H₁: r ¹ O, reject H₀ if the absolute value of r is greater than the critical value in the table.

3. The heights (in inches) and pulse rates (in beats per minute) for a sample of 17 women were measured. Using technology with the paired height/pulse data, the linear correlation coefficient is found to be 0.654. Is there sufficient evidence to support the claim that there is a linear correlation between the heights and pulse rates of women? Use a significance level of a = 0.01.

Click here to view a table of critical values for the correlation coefficient.²

Because |0.654| is (1) ________ than the critical value, there (2) _______ sufficient evidence to support the claim that there is a linear correlation between the heights and pulse rates of women for a significance level of a = 0.01.

2: Table of Critical Values

n	a = 0.5	a = .01
4	.950	.990
5	.878	.959
6	.811	.917
7	.754	.875
8	.707	.834
9	.666	.798
10	.632	.765
11	.602	.735
12	.576	.708
13	.553	.684
14	.532	.661
15	.514	.641
16	.497	.623
17	.482	.606
18	.468	.590
19	.456	.575
20	.444	.561
25	.396	.505
30	.361	.463
35	.335	.430
40	.312	.402
45	.294	.378
50	.279	.361
60	.254	.330
70	.236	.305
80	.220	.286
90	.207	.269
100	.196	.256

 

NOTE: To test H₀: r = O against H₁: r ¹ O, reject H₀ if the absolute value of r is greater than the critical value in the table.

(1) a) greater         b) less

(2) a) is                  b) is not

4. Use the given data set to complete parts (a) through (c) below. (Use a = 0.05.)

x	10	8	13	9	11	14	6	4	12	7	5
y	9.14	8.14	8.73	8.76	9.27	8.09	6.14	3.11	9.13	7.26	4.75

 

³Click here to view a table of critical values for the correlation coefficient.

a. Construct a scatterplot. Choose the correct graph below.

b. Find the linear correlation coefficient, r, then determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables.

The linear correlation coefficient is r = ______ .

(Round to three decimal places as needed.)

Using the linear correlation coefficient found in the previous step, determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables. Choose the correct answer below.

A. There is insufficient evidence to support the claim of a linear correlation between the two variables.

B. There is sufficient evidence to support the claim of a nonlinear correlation between the two variables.

C. There is sufficient evidence to support the claim of a linear correlation between the two variables.

D. There is insufficient evidence to support the claim of a nonlinear correlation between the two variables.

c. Identify the feature of the data that would be missed if part (6) was completed without constructing the scatterplot.

Choose the correct answer below.

A. The scatterplot reveals a distinct pattern that is a straight-line pattern with positive slope.

B. The scatterplot reveals a distinct pattern that is not a straight-line pattern.

C. The scatterplot reveals a distinct pattern that is a straight-line pattern with negative slope.

D. The scatterplot does not reveal a distinct pattern.

3: Table of Critical Values

n	a = 0.5	a = .01
4	.950	.990
5	.878	.959
6	.811	.917
7	.754	.875
8	.707	.834
9	.666	.798
10	.632	.765
11	.602	.735
12	.576	.708
13	.553	.684
14	.532	.661
15	.514	.641
16	.497	.623
17	.482	.606
18	.468	.590
19	.456	.575
20	.444	.561
25	.396	.505
30	.361	.463
35	.335	.430
40	.312	.402
45	.294	.378
50	.279	.361
60	.254	.330
70	.236	.305
80	.220	.286
90	.207	.269
100	.196	.256

 

NOTE: To test H₀: r = O against H₁: r ¹ O, reject H₀ if the absolute value of r is greater than the critical value in the table.

5. Use the given data set to complete parts (a) through (c) below. (Use a = 0.05.)

x	10	8	13	9	11	14	6	4	12	7	5
y	7.46	6.77	12.75	7.11	7.81	8.84	6.08	5.39	8.15	6.43	5.73

 

⁴Click here to view a table of critical values for the correlation coefficient.

a. Construct a scatterplot. Choose the correct graph below.

b. Find the linear correlation coefficient, r, then determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables.

The linear correlation coefficient is r = ________.

(Round to three decimal places as needed.)

Using the linear correlation coefficient found in the previous step, determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables. Choose the correct answer below.

A. There is sufficient evidence to support the claim of a nonlinear correlation between the two variables.

B. There is insufficient evidence to support the claim of a nonlinear correlation between the two variables.

C. There is insufficient evidence to support the claim of a linear correlation between the two variables.

D. There is sufficient evidence to support the claim of a linear correlation between the two variables.

c. Identify the feature of the data that would be missed if part (6) was completed without constructing the scatterplot.

Choose the correct answer below.

A. The scatterplot reveals a perfect straight-line pattern, except for the presence of one outlier.

B. The scatterplot does not reveal a perfect straight-line pattern.

C. The scatterplot does not reveal a perfect straight-line pattern, and contains one outlier.

D. The scatterplot reveals a perfect straight-line pattern and does not contain any outliers.

4: Table of Critical Values

n	a = 0.5	a = .01
4	.950	.990
5	.878	.959
6	.811	.917
7	.754	.875
8	.707	.834
9	.666	.798
10	.632	.765
11	.602	.735
12	.576	.708
13	.553	.684
14	.532	.661
15	.514	.641
16	.497	.623
17	.482	.606
18	.468	.590
19	.456	.575
20	.444	.561
25	.396	.505
30	.361	.463
35	.335	.430
40	.312	.402
45	.294	.378
50	.279	.361
60	.254	.330
70	.236	.305
80	.220	.286
90	.207	.269
100	.196	.256

 

NOTE: To test H₀: r = O against H₁: r ¹ O, reject H₀ if the absolute value of r is greater than the critical value in the table.

6. Refer to the accompanying scatterplot. The four points in the lower left corner are measurements from women, and the four points in the upper right corner are from men.

Complete parts (a) through (e) below.

a. Examine the pattern of the four points in the lower left corner (from women) only, and subjectively determine whether there appears to be a correlation between x and y for women. Choose the correct answer below.

A. There appears to be a linear correlation because the points form an obvious pattern.

B. There does not appear to be a linear correlation because the points do not form a line.

C. There does not appear to be a linear correlation because the points form an obvious pattern.

D. There appears to be a linear correlation because the points form a line.

b. Examine the pattern of the four points in the upper right corner (from men) only, and subjectively determine whether there appears to be a correlation between x and y for men. Choose the correct answer below.

A. There appears to be a linear correlation because the points form an obvious pattern.

B. There appears to be a linear correlation because the points form a line.

C. There does not appear to be a linear correlation because the points form an obvious pattern.

D. There does not appear to be a linear correlation because the points do not form a line.

c. Find the linear correlation coefficient using only the four points in the lower left corner (for women). Will the four points in the upper right corner (for men) have the same linear correlation coefficient?

The correlation coefficient for the points in the lower left corner is r = ______ .

(Type an integer or a fraction.)

Do the four points in the upper right corner have the same correlation coefficient?

A. Yes, because the four points in the upper right corner form a different pattern from the four points in the lower left corner.

B. Yes, because the four points in the upper right corner form the same pattern as the four points in the lower left corner.

C. No, because the four points in the upper right corner form the same pattern as the four points in the lower left corner.

D. No, because the four points in the upper right corner form a different pattern from the four points in the lower left corner.

d. Find the value of the linear correlation coefficient using all eight points. What does that value suggest about the relationship between x and y? Use a = 0.05.

The correlation coefficient for all eight points is r = _____.

(Round to three decimal places as needed.)

Using a = 0.05, what does r suggest about the relationship between x and y?

A. There is not sufficient evidence to support the claim of a linear correlation, because the correlation coefficient is greater than the critical value.

B. There is not sufficient evidence to support the claim of a linear correlation, because the correlation coefficient is less than the critical value.

C. There is sufficient evidence to support the claim of a linear correlation, because the correlation coefficient is less than the critical value.

D. There is sufficient evidence to support the claim of a linear correlation, because the correlation coefficient is greater than the critical value.

e. Based on the preceding results, what can be concluded? Should the data from women and the data from men be considered together, or do they appear to represent two different and distinct populations that should be analysed separately?

 A. There are two different populations that should be considered separately.

B. There are two different populations that should be considered together.

C. The two data sets are the same population and they should be considered separately.

D. The two data sets are the same population and they should be considered together.

7. Listed below are annual data for various years. The data are weights (metric tons) of imported lemons and car crash fatality rates per 100,000 population. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using a = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between lemon imports and crash fatality rates? Do the results suggest that imported lemons cause car fatalities?

Lemon Imports	232	264	359	483	530
Crash Fatality Rate	15.9	15.7	15.4	15.2	14.8

 

What are the null and alternative hypotheses?

A. H₀: r = 0

     H₁: r ¹ 0

B. H₀: r = 0

    H₁: r < 0

C. H₀: r ¹ 0

     H₁: r = 0

D. H₀: r = 0

     H₁: r > 0

Construct a scatterplot. Choose the correct graph below.

The linear correlation coefficient is r = ______.

(Round to three decimal places as needed. )

The test statistic is t = __________ .

(Round to three decimal places as needed. )

The P-value is _______.

(Round to three decimal places as needed. )

Because the P-value is (1) ______ than the Significance level 0.05, there (2) ______ sufficient evidence to support the claim that there is a linear correlation between lemon imports and crash fatality rates for a significance level of a = 0.05.

Do the results suggest that imported lemons cause car fatalities?

A. The results suggest that an increase in imported lemons causes in an increase in car fatality rates.

B. The results do not suggest any cause-effect relationship between the two variables.

C. The results suggest that an increase in imported lemons causes car fatality rates to remain the same.

D. The results suggest that imported lemons cause car fatalities.

(1) a) less     b) greater

(2) a) is     b) is not

8. Listed below are the budgets (in millions of dollars) and the gross receipts (in millions of dollars) for randomly selected movies. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using a = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between budgets and gross receipts? Do the results change if the actual budgets listed are $62,000,000, $91,000,000, $52,000,000, and so on?

Budget (x)	62	91	52	34	208	98	88
Gross (y)	65	62	44	52	507	140	42

 

What are the null and alternative hypotheses?

A. H₀: r = 0

     H₁: r < 0

B. H₀: r ¹ 0

    H₁: r = 0

C. H₀: r = 0

     H₁: r > 0

D. H₀: r = 0

     H₁: r ¹ 0

Construct a scatterplot. Choose the correct graph below.

(A. () B. (C, (D.

 

The linear correlation coefficient r is _______.

(Round to three decimal places as needed. )

The test statistic t is __________.

(Round to two decimal places as needed.)

The P-value is _____________.

(Round to three decimal places as needed. )

Because the P-value is (1) _________ than the significance level 0.05, there (2) _____ sufficient evidence to support the claim that there is a linear correlation between between budgets and gross receipts for a significance level of a = 0.05.

Do the results change if the actual budgets listed are $62,000,000, $91,000,000, $52,000,000, and so on?

A. Yes, the results would change because it would result in a different linear correlation coefficient.

B. No, the results do not change because it would result in the same linear correlation coefficient.

C. Yes, the results would need to be multiplied by 1,000,000.

D. No, the results do not change because it would result in a different linear correlation coefficient.

(1) a) greater     b) less

(2) a) is        b) is not

9. Listed below are the overhead widths (in cm) of seals measured from photographs and the weights (in kg) of the seals. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the critical values of r using a = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between overhead widths of seals from photographs and the weights of the seals?

Overhead Width	7.1	7.5	9.8	9.3	8.6	8.3
Weight	115	176	246	200	197	190

 

Click here to view a table of critical values for the correlation coefficient.⁵

Construct a scatterplot. Choose the correct graph below.

(A. () B. (3C, (3D.

weight (kg) weight (kg) weight (kg) weight (kg)

width (cm) width (cm) width (cm) width (cm)

The linear correlation coefficient is r = _______.

(Round to three decimal places as needed. )

The critical values are r = ___________.

(Round to three decimal places as needed. Use a comma to separate answers as needed.)

Because the absolute value of the linear correlation coefficient is (1) ________ than the positive critical value, there (2) _____ sufficient evidence to support the claim that there is a linear correlation between overhead widths of seals from photographs and the weights of the seals for a significance level of a = 0.05.

5: Table of Critical Values

n	a = 0.5	a = .01
4	.950	.990
5	.878	.959
6	.811	.917
7	.754	.875
8	.707	.834
9	.666	.798
10	.632	.765
11	.602	.735
12	.576	.708
13	.553	.684
14	.532	.661
15	.514	.641
16	.497	.623
17	.482	.606
18	.468	.590
19	.456	.575
20	.444	.561
25	.396	.505
30	.361	.463
35	.335	.430
40	.312	.402
45	.294	.378
50	.279	.361
60	.254	.330
70	.236	.305
80	.220	.286
90	.207	.269
100	.196	.256

 

NOTE: To test H₀: r = O against H₁: r ¹ O, reject H₀ if the absolute value of r is greater than the critical value in the table.

(1) a) greater    b) less

(2) a) is      b) is not

10. The data below shows the high temperatures and the times (in minutes) runners who won a marathon. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using a = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between temperature and winning times? What are the null and alternative hypotheses?

Temperature (x)	58	62	49	63	75	70	52	57
Time (y)	145.616	147.407	147.556	145.043	146.673	146.405	144.405	147.274

 

What are the null and alternative hypotheses?

A. H₀: r ¹ 0

     H₁: r = 0

B. H₀: r = 0

    H₁: r ¹ 0

C. H₀: r = 0

     H₁: r < 0

D. H₀: r = 0

     H₁: r > 0

Construct the scatterplot. Choose the correct graph below.

The linear correlation coefficient r is ______.

(Round to three decimal places as needed. )

The test statistic t is ________.

(Round to three decimal places as needed. )

The P-value is _______.

(Round to three decimal places as needed. )

Because the P-value is (1) ______ than the significance level 0.05, there (2) ______ sufficient evidence to support the claim that there is a linear correlation between between temperature and winning times for a significance level of a =0.05.

Does it appear that winning times are affected by temperature?

Yes, because there is not a linear correlation between the two variables.

No, because there is not a linear correlation between the two variables.

No, because there is a linear correlation between the two variables.

Yes, because there is a linear correlation between the two variables.

(1) a) greater     b) less

(2) a) is          b) is not

11. The data below shows the selling price (in hundred thousands) and the list price (in hundred thousands) of homes sold. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using a = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between the two variables?

Selling Price (x)	404	300	379	430	455	475	319	350	417	335
List Price (y)	414	316	390	436	488	476	323	370	433	345

 

What are the null and alternative hypotheses?

A. H₀: r = 0

     H₁: r ¹ 0

B. H₀: r = 0

    H₁: r < 0

C. H₀: r ¹ 0

     H₁: r = 0

D. H₀: r = 0

     H₁: r > 0

Construct a scatterplot. Choose the correct graph below.

The linear correlation coefficient r is ________.

(Round to three decimal places as needed. )

The test statistic t is ________.

(Round to three decimal places as needed. )

The P-value is ______.

(Round to three decimal places as needed. )

Because the P-value is (1) _______ than the significance level 0.05, there (2)______ sufficient evidence to support the claim that there is a linear correlation between selling price (in hundred thousands) and the list price (in hundred thousands) of homes sold for a significance level of a = 0.05.

(1) a) less           b) greater

(2) a) is              b) is not

12. The data below shows the annual salaries (in millions) and the number of viewers (in millions) of eight television actors and actresses. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using a = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between the two variables?

Salary (x)	92	15	12	31	11	7	8	2
Viewers (Y)	17	4.2	5.8	1.9	10.8	9.1	13.2	4.6

 

What are the null and alternative hypotheses?

A. H₀: r ¹ 0

     H₁: r = 0

B. H₀: r = 0

    H₁: r ¹ 0

C. H₀: r = 0

     H₁: r > 0

D. H₀: r = 0

     H₁: r < 0

Construct a scatterplot. Choose the correct graph below.

The linear correlation coefficient r is _______.

(Round to three decimal places as needed. )

The test statistic t is _____________.

(Round to three decimal places as needed. )

The P-value is ______.

(Round to three decimal places as needed. )

Because the P-value is (1) _______ than the significance level 0.05, there (2) _______ sufficient evidence to support the claim that there is a linear correlation between annual salaries (in millions) and the number of viewers (in millions) for a significance level of a = 0.05.

Can the number of viewers be used to get a good sense of annual salaries?

A. Knowing the number of viewers is not helpful in getting a good sense for the annual salaries because there does not appear to be a linear correlation between the two variables.

B. Knowing the number of viewers is helpful in getting a good sense for the annual salaries because there appears to be a linear correlation between the two variables.

C. Knowing the number of viewers is not helpful in getting a good sense for the annual salaries because there appears to be a linear correlation between the two variables.

D. Knowing the number of viewers is helpful in getting a good sense for the annual salaries, because there does not appear to be a linear correlation between the two variables.

(1) a) less      b) greater

(2) a) is         b) is not

13. The data below shows height (in inches) and pulse rates (in beats per minute) of a random sample of women. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using a = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between height and pulse rate?

Full data set

Height (x)	65.6	65.8	61.3	62.4	59.8	65.3	60.4	64.9	67.9	60.8
Pulse rate (y)	76	75	86	63	71	66	82	65	67	67

 

Height (x)	67.1	65.7	58.3	61.8	58.5	63.6	70.5	59.9	69.7	61.2
Pulse rate (Y)	82	76	71	71	72	72	77	79	74	75

 

What are the null and alternative hypotheses?

A. H₀: r = 0

     H₁: r < 0

B. H₀: r = 0

    H₁: r > 0

C. H₀: r ¹ 0

     H₁: r = 0

D. H₀: r = 0

     H₁: r ¹ 0

 

Construct a scatterplot. Choose the correct graph below.

The linear correlation coefficient r is __________.

(Round to three decimal places as needed. )

The test statistic t is _____________________.

(Round to three decimal places as needed. )

The P-value is __________________.

(Round to three decimal places as needed. )

Is there sufficient evidence to conclude that there is a linear correlation between the two variables?

A. Yes, because the P-value is less than the significance level.

B. No, because the P-value is greater than the significance level.

C. No, because the P-value is less than the significance level.

D. Yes, because the P-value is greater than the significance level.

14. Fill in the blank.

A__________ exists between two variables when the values of one variable are somehow associated with the values of the other variable.

A (1) ___sexists between two variables when the values of one variable are somehow associated with the values of the other variable.

(1) (a) trial

(b) difference

(c) inference

(d) correlation

15. Fill in the blank.

When determining whether there is a correlation between two variables, one should use a ___ to explore the data visually.

When determining whether there is a correlation between two variables, one should use a (1) _____ to explore the data visually.

(1) a) protractor

b) probability distribution

(c) scatterplot

d) correlation coefficient

16. Which of the following statements about correlation is true?

Choose the correct answer below.

A. We say that there is a negative correlation between x and y if the x-values increase as the corresponding y-values increase.

B. We say that there is a positive correlation between x and y if there is no distinct pattern in the scatterplot.

C. We say that there is a positive correlation between x and y if the x-values increase as the corresponding y-values increase.

D. We say that there is a positive correlation between x and y if the x-values increase as the corresponding y-values decrease.

17. Fill in the blank.

The _____ measures the strength of the linear correlation between the paired quantitative x- and y-values in a sample.

The (1) ___ measures the strength of the linear correlation between the paired quantitative x- and y-values in a sample.

(1) (a) population mean m

(b) test statistic t

(c) sample standard deviation s

(d) linear correlation coefficient r

18. Which of the following is NOT a requirement in determining whether there is a linear correlation between two variables?

Choose the correct answer below.

A. A scatterplot should visually show a straight-line pattern.

B. The sample of paired data is a simple random sample of quantitative data.

C. If r>1, then there is a positive linear correlation.

D. Any outliers must be removed if they are known to be errors.

19. Which of the following is NOT a property of the linear correlation coefficient r?

Choose the correct answer below.

A. The value of r is not affected by the choice of x or y.

B. The value of r is always between - 1 and 1 inclusive.

C. The value of r measures the strength of a linear relationship.

D. The linear correlation coefficient r is robust. That is, a single outlier will not affect the value of r.

20. Which of the following is NOT one of the three common errors involving correlation?

Choose the correct answer below.

A. The use of data based on averages

B. Correlation does not imply causality

C. Mistaking no linear correlation with no correlation

D. The conclusion that correlation implies causality

21. Which of the following is NOT true for a hypothesis test for correlation?

Choose the correct answer below.

A. If the P-value is less than or equal to the significance level, we should reject the null hypothesis and conclude that there is sufficient evidence to support the claim of a linear correlation.

B. If |r| > critical value, we should fail to reject the null hypothesis and conclude that there is not sufficient evidence to support the claim of a linear correlation.

C. If the P-value is greater than the significance level, we should fail to reject the null hypothesis and conclude that there is not sufficient evidence to support the claim of a linear correlation.

D. If |r| £ critical value, we should fail to reject the null hypothesis and conclude that there is not sufficient evidence to support the claim of a linear correlation.