In: Statistics and Probability
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?
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.