Question

Confidence intervals and margin of error.

	A	B	C	D	E	F
1	County	Average Income	Year	County	Average Income	Year
2	A-1	$42,944	2014	A-1	$51,500	2018
3	B-1	$48,795	2014	B-1	$51,300	2018
4	C-1	$53,475	2014	C-1	$59,800	2018
5	D-1	$48,692	2014	D-1	$47,200	2018
6	E-1	$38,927	2014	E-1	$42,600	2018
7	F-1	$69,919	2014	F-1	$80,600	2018
8	G-1	$49,837	2014	G-1	$56,300	2018
9	H-1	$51,038	2014	H-1	$55,700	2018
10	I-1	$40,809	2014	I-1	$46,300	2018
11	J-1	$49,164	2014	J-1	$51,800	2018
12	K-1	$44,230	2014	K-1	$50,200	2018
13	L-1	$47,748	2014	L-1	$53,900	2018
14	M-1	$39,534	2014	M-1	$46,200	2018
15	N-1	$44,689	2014	N-1	$49,200	2018
16	O-1	$57,465	2014	O-1	$63,900	2018
17	P-1	$47,387	2014	P-1	$52,000	2018
18	Q-1	$48,116	2014	Q-1	$52,600	2018
19	R-1	$37,339	2014	R-1	$41,400	2018
20	S-1	$53,814	2014	S-1	$62,900	2018
21	T-1	$45,806	2014	T-1	$54,600	2018
22	U-1	$39,998	2014	U-1	$44,700	2018
23	V-1	$53,367	2014	V-1	$58,700	2018
24	W-1	$47,905	2014	W-1	$43,600	2018
25	X-1	$45,992	2014	X-1	$57,700	2018
26	Y-1	$44,300	2014	Y-1	$46,700	2018
27	Z-1	$49,320	2014	Z-1	$54,700	2018
28	AA-1	$39,151	2014	AA-1	$40,200	2018
29	AB-1	$42,684	2014	AB-1	$46,000	2018
30	AC-1	$88,429	2014	AC-1	$92,400	2018
31	AD-1	$66,131	2014	AD-1	$70,700	2018
32	AE-1	$50,704	2014	AE-1	$57,600	2018
33	AF-1	$70,886	2014	AF-1	$78,500	2018
34	AG-1	$41,565	2014	AG-1	$45,400	2018
35	AH-1	$42,148	2014	AH-1	$49,200	2018
36	AI-1	$43,552	2014	AI-1	$50,100	2018
37	AJ-1	$46,573	2014	AJ-1	$51,900	2018
38	AK-1	$54,092	2014	AK-1	$56,000	2018
39	AL-1	$42,410	2014	AL-1	$46,900	2018
40	AM-1	$43,104	2014	AM-1	$51,500	2018
41	AN-1	$46,023	2014	AN-1	$50,400	2018
42	AO-1	$61,671	2014	AO-1	$67,200	2018
43	AP-1	$41,356	2014	AP-1	$43,900	2018
44	AQ-1	$50,153	2014	AQ-1	$56,000	2018
45	AR-1	$47,870	2014	AR-1	$60,400	2018
46	AS-1	$48,015	2014	AS-1	$53,600	2018
47	AT-1	$46,127	2014	AT-1	$52,800	2018
48	AU-1	$39,447	2014	AU-1	$48,000	2018
49	AV-1	$42,707	2014	AV-1	$45,900	2018
50	AW-1	$41,586	2014	AW-1	$45,300	2018
51	AX-1	$46,129	2014	AX-1	$53,800	2018
52	AY-1	$44,597	2014	AY-1	$51,000	2018
53	AZ-1	$43,608	2014	AZ-1	$47,700	2018
54	BA-1	$39,309	2014	BA-1	$44,400	2018
55	BB-1	$45,404	2014	BB-1	$51,500	2018
56	BC-1	$52,886	2014	BC-1	$60,500	2018
57	BD-1	$50,129	2014	BD-1	$51,200	2018
58	BE-1	$44,192	2014	BE-1	$52,300	2018
59	BF-1	$49,301	2014	BF-1	$54,400	2018
60	BG-1	$38,651	2014	BG-1	$41,100	2018
61	BH-1	$41,106	2014	BH-1	$46,100	2018
62	BI-1	$41,311	2014	BI-1	$47,600	2018
63	BJ-1	$42,942	2014	BJ-1	$48,900	2018
64	BK-1	$44,977	2014	BK-1	$49,600	2018
65	BL-1	$63,879	2014	BL-1	$67,300	2018
66	BM-1	$61,587	2014	BM-1	$61,600	2018
67	BN-1	$41,818	2014	BN-1	$47,200	2018
68	BO-1	$47,161	2014	BO-1	$50,900	2018
69	BP-1	$40,656	2014	BP-1	$46,800	2018
70	BQ-1	$48,227	2014	BQ-1	$55,100	2018
71	BR-1	$44,979	2014	BR-1	$50,400	2018
72	BS-1	$40,299	2014	BS-1	$47,700	2018
73	BT-1	$49,965	2014	BT-1	$53,000	2018
74	BU-1	$55,824	2014	BU-1	$58,200	2018
75	BV-1	$45,248	2014	BV-1	$49,000	2018
76	BW-1	$41,700	2014	BW-1	$46,600	2018
77	BX-1	$46,412	2014	BX-1	$54,100	2018
78	BY-1	$41,223	2014	BY-1	$46,700	2018
79	BZ-1	$42,497	2014	BZ-1	$45,300	2018
80	CA-1	$43,529	2014	CA-1	$51,400	2018
81	CB-1	$53,917	2014	CB-1	$60,800	2018
82	CC-1	$44,796	2014	CC-1	$46,900	2018
83	CD-1	$41,784	2014	CD-1	$46,800	2018
84	CE-1	$42,709	2014	CE-1	$51,400	2018
85	CF-1	$40,091	2014	CF-1	$43,600	2018
86	CG-1	$45,849	2014	CG-1	$51,600	2018
87	CH-1	$53,781	2014	CH-1	$55,500	2018
88	CI-1	$60,551	2014	CI-1	$65,800	2018
89	CJ-1	$41,598	2014	CJ-1	$48,900	2018
90	CK-1	$35,220	2014	CK-1	$43,000	2018
91	CL-1	$49,264	2014	CL-1	$56,100	2018
92	CM-1	$48,274	2014	CM-1	$47,400	2018
93	CN-1	$52,081	2014	CN-1	$54,900	2018

Use Excel to calculate the total average income for all counties for both years, the standard deviation for both years, sample size for both years, confidence coefficient for both years, margin of error for both years, and correlation coefficient for both years. Then, compare the two years and explain your findings in at least two paragraphs. Does anything stand out? If there were changes, what could you attribute them to? Do you notice anything questionable about the data? If so, what?

Answer 1

Using excel the following excel functions to compute the required parameters:

Averages:

We may drag the cell to the next to obtain the average for the year 2018.

Standard Deviations:

Sample size:

Suppose we wish to construct a 95% CI for the difference in means (or) we wish to test the equality of two means at 5% level of significance.

Then the confidence coefficient would be 0.95. To obtain the margin of error, since the population standard deviation is unknown, we must construct the interval using the t statistic, instead of Z.

And correlation computed using the function:

We get:

	2014	2018
Average	$47,461	$52,610
SD	8162.79661	8525.309
Sample size	92	92
Conf. Coefficient	0.95	0.95
Margin of Error	1690.47	1765.54
Correlation	0.95

From the figures obtained above,

We find that the average income has increased over the years (from 2014 to 2018). Also, a greater variation is observed in the incomes recorded in 2018 as compared to that in 2014. Now, in order to determine whether this difference in variation and average is statistically significant, we may run the following tests on the given data:

To test the equality of variance, where we test:

Vs       

The appropriate statistical test here would be F test for equality of variance; Using excel:

We find that the test is not significant at 5% level (since p-value = 0.340 > 0.05). We may conclude that the variation in income in 2018 although greater than that in 2014, the difference between the two is not statistically significant.

Now to test whether the two means/averages are significantly different:

Vs

We find that the test is significant at 5% (p-value = 0.000 < 0.05). We may say that the difference in the average income recorded in 2014 and 2018 is statistically significant. We observe that the average income has significantly increased over the years. And this increase can be attributed to several socio-economic factors and not merely due to chance.

Also, if we compare the confidence intervals obtained for the years 2014 and 2018

(49151.47, 45770.53) and (54375.54, 50844.46), the former is narrower and hence a better interval estimate among the two.

Regarding the correlation, between the two, we get the Pearsons correlation coefficient r = 0.95. However, since, the observations are on the same phenomena - Income, recorded at two different intervals, we are already aware of the strong dependency/relation between the two. Hence, the coefficient here, would make no practical sense in this case.