IMathAS-libretexts

To best answer our original question, it might make the most sense to test for significant correlation between income and drawn nickel size. Incomes (in thousands of $) are shown for each of the 75 samples below:

Income (thousands of $)	Coin size (mm)
10	16
28	16
35	26
34	22
31	16
8	19
30	18
36	19
25	25
34	17
21	18
20	21
12	27
17	21
18	24
8	18
12	29
23	26
17	12
39	20
14	22
26	22
24	28
10	30
39	30
34	19
26	19
35	21
26	17
11	20
23	24
21	17
25	16
21	21
29	24
22	25
39	21
18	27
33	26
20	15
71	17
43	19
66	18
92	27
91	21
47	20
79	19
57	19
46	22
94	21
74	18
100	18
56	24
78	25
60	19
61	18
90	19
58	22
87	25
81	18
48	20
47	13
87	21
52	21
77	24
48	15
42	22
89	18
51	12
74	17
51	16
62	16
56	21
41	17
99	17

You can copy the data into Excel by highlighting the data, right-clicking and selecting Copy, then opening Excel, clicking on a blank cell, and selecting Paste from the Edit menu.

Test the claim that there is significant correlation at the 0.05 significance level. Retain at least 3 decimals on all values.

a) If we use

\displaystyle {L}

to denote the low income group and

\displaystyle {H}

to denote the high income group, identify the correct alternative hypothesis.

$\displaystyle {H}_{{1}}:{r}\ne{0}$
$\displaystyle {H}_{{1}}:\rho={0}$
$\displaystyle {H}_{{1}}:{p}_{{L}}\ne{p}_{{H}}$
$\displaystyle {H}_{{1}}:\rho\ne{0}$
$\displaystyle {H}_{{1}}:\mu\ne{0}$

b) The

\displaystyle {r}

test statistic value is:
Hint: You may find it more convenient to use Excel's CORREL, SLOPE, and INTERCEPT functions rather than your calculator

c) The critical value is:
If your sample size falls between table values, use the smaller sample size

d) Based on this, we

Reject $\displaystyle {H}_{{0}}$
Fail to reject $\displaystyle {H}_{{0}}$

e) Which means

The sample data supports the claim
There is not sufficient evidence to support the claim
There is sufficient evidence to warrant rejection of the claim
There is not sufficient evidence to warrant rejection of the claim

f) The regression equation (in terms of income

\displaystyle {x}

) is:

\displaystyle \hat{{y}}=

g) To predict what diameter a child would draw a nickel given family income, it would be most appropriate to:

Use the regression equation
Use the centroid
Use the P-Value
Use the residual

License

\displaystyle

Income (thousands of $)	Coin size (mm)
10	16
28	16
35	26
34	22
31	16
8	19
30	18
36	19
25	25
34	17
21	18
20	21
12	27
17	21
18	24
8	18
12	29
23	26
17	12
39	20
14	22
26	22
24	28
10	30
39	30
34	19
26	19
35	21
26	17
11	20
23	24
21	17
25	16
21	21
29	24
22	25
39	21
18	27
33	26
20	15
71	17
43	19
66	18
92	27
91	21
47	20
79	19
57	19
46	22
94	21
74	18
100	18
56	24
78	25
60	19
61	18
90	19
58	22
87	25
81	18
48	20
47	13
87	21
52	21
77	24
48	15
42	22
89	18
51	12
74	17
51	16
62	16
56	21
41	17
99	17

Income (thousands of $)	Coin size (mm)
10	16
28	16
35	26
34	22
31	16
8	19
30	18
36	19
25	25
34	17
21	18
20	21
12	27
17	21
18	24
8	18
12	29
23	26
17	12
39	20
14	22
26	22
24	28
10	30
39	30
34	19
26	19
35	21
26	17
11	20
23	24
21	17
25	16
21	21
29	24
22	25
39	21
18	27
33	26
20	15
71	17
43	19
66	18
92	27
91	21
47	20
79	19
57	19
46	22
94	21
74	18
100	18
56	24
78	25
60	19
61	18
90	19
58	22
87	25
81	18
48	20
47	13
87	21
52	21
77	24
48	15
42	22
89	18
51	12
74	17
51	16
62	16
56	21
41	17
99	17

Income (thousands of $)	Coin size (mm)
10	16
28	16
35	26
34	22
31	16
8	19
30	18
36	19
25	25
34	17
21	18
20	21
12	27
17	21
18	24
8	18
12	29
23	26
17	12
39	20
14	22
26	22
24	28
10	30
39	30
34	19
26	19
35	21
26	17
11	20
23	24
21	17
25	16
21	21
29	24
22	25
39	21
18	27
33	26
20	15
71	17
43	19
66	18
92	27
91	21
47	20
79	19
57	19
46	22
94	21
74	18
100	18
56	24
78	25
60	19
61	18
90	19
58	22
87	25
81	18
48	20
47	13
87	21
52	21
77	24
48	15
42	22
89	18
51	12
74	17
51	16
62	16
56	21
41	17
99	17