Question

In: Statistics and Probability

Confidence intervals and margin of error. A B C D E F 1 County Average Income...

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?

Solutions

Expert Solution

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.


Related Solutions

Confidence Intervals (Proportions) 1 Find the margin of error and 95% confidence interval for the following...
Confidence Intervals (Proportions) 1 Find the margin of error and 95% confidence interval for the following surveys. Round all answers to 2 decimal places. (a) A survey of 500 people finds that 56% plan to vote for Smith for governor. Margin of Error (as a percentage): Confidence Interval: % to % (b) A survey of 1500 people finds that 47% support stricter penalties for child abuse. Margin of Error (as a percentage): Confidence Interval: % to % 2 Assume that...
Consider the cross: A/a; b/b; C/c; D/d; E/e x A/a; B/b; c/c; D/d; e/e a) what...
Consider the cross: A/a; b/b; C/c; D/d; E/e x A/a; B/b; c/c; D/d; e/e a) what proportion of the progeny will phenotypically resemble the first parent? b) what proportion of the progeny will genotypically resemble neither parent?
Find the 80%, 95%, and 99% confidence intervals & margin error Create your own confidence interval...
Find the 80%, 95%, and 99% confidence intervals & margin error Create your own confidence interval (you cannot use 80%, 95%, and 99%) and make sure to show your work. Make sure to list the margin of error. 122 132 97 102 106 116 94 132 112 123 123 108 84 120 125 123 115 103 127 139 122 124 112 113 109 104 120 109 117 108 125 109 119 138 125 104 110 101 130 124 115 104...
Find the mean, standard deviation, 80%, 95%, and 99% confidence intervals, and margin of error for...
Find the mean, standard deviation, 80%, 95%, and 99% confidence intervals, and margin of error for those intervals. Show all work. $32,096 $29,716 $29,227 $27,209 $22,794 $19,306 $19,238 $19,164 18,994 $18,329 $17,978 $17,626 $16,823 $16,709 $15,967 $15,921 $15,908 $15,905 $15,638 $15,629 $14,414 $14,229 $14,137 $14,051 $13,907 $13,776 $13,281 $12,739 $12,300 $11,994 $11,262 $11,112 $10,984 $10,824 $10,823 $10,448 $10,081 $9,883 $9,668 $9,663 $9,337 $8,855 $8,741 $8,596 $8,385 $7,204 $7,072 $6,503 $5,653 $4,602
If there are 7 total notes C, D, E, F, G, A, and B and if...
If there are 7 total notes C, D, E, F, G, A, and B and if a five-note melody is selected at random (so that all melodies counted in part (a) are equally likely to be chosen), what is the probability that the melody will include exactly two “A” notes, but no other repeated notes? (A few allowable examples: AACEG, ACAEG, DFACA, EAABC, etc.)
Seven people (A,B,C,D,E, F, and G) are seated in a row. Suppose A,B, and C are...
Seven people (A,B,C,D,E, F, and G) are seated in a row. Suppose A,B, and C are freshmen, D and E are sophomores and F and G are juniors. How many arrangements are possible if: (a) D and F must sit together? (b) A and C must not sit together? (c) All freshmen must sit together? (d) All freshmen must sit together, all sophomores must sit together, and all juniors must sit together? (e) Exactly two people sit between A and...
1. a. How many permutations are there of the letters {A,B,C,D,E,F}? Of there, how many are...
1. a. How many permutations are there of the letters {A,B,C,D,E,F}? Of there, how many are even? b. Express the permutation BAFEDC in P6 in cycle notation and determine whether it is even or odd. c. Determine the composition BAFEDC*BCAFDE in P6. Is the composition even or odd? d. What is the members of P6 whose cycle notation (1345)(26)?
Six people (A, B, C, D, E, F) are lining up to enter a museum. (1)...
Six people (A, B, C, D, E, F) are lining up to enter a museum. (1) How many ways are there to line up the six people if A is somewhere before B, although not necessarily immediately before? (2) How many ways are there to line up the six people if A is next to C? (3) How many ways are there to line up the six people if A must be next to B but C and D can...
Find 75%, 80%, 95%, and 99% confidence intervals, please include the margin of error.   4,2,4,2,3,1,3,0,4,3,6,4,7,11,14,14,4,6,2,12,2,3,3,11,4,3,4,7,3,1,1,2,6,7,0,5,3,1,1,7,5,0,6,2,3,5,4,1,3,1
Find 75%, 80%, 95%, and 99% confidence intervals, please include the margin of error.   4,2,4,2,3,1,3,0,4,3,6,4,7,11,14,14,4,6,2,12,2,3,3,11,4,3,4,7,3,1,1,2,6,7,0,5,3,1,1,7,5,0,6,2,3,5,4,1,3,1
Assume that: float a, b, c, d, f; and variables b, c, d, f are initialized....
Assume that: float a, b, c, d, f; and variables b, c, d, f are initialized. Write a line of c++ code that calculates the formula below and stores the result to the variable a:
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT