42. Show that in Example 4.4.7, page 153, the difference between the two candidates is statistically significant even at a confidence level of 99.7%.

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Example 4.4.7. One week before the election a new public poll is taken.
In the two weeks between the polls, Candidate B increases his support among
likely voters. The results of the new poll are 581 in favor of Candidate A, and
769 in favor of Candidate B. Is the difference now statistically significant?
CHAPTER 4. STATISTICS
154
The sample size is n = 1350, and the poll presents only two options, so
there is a common standard error.
581
For Candidate A, we have k = 581, so
2 0.4303 or 43.03%. For
1350
769
Candidate B, we have k = 769, so p =
2 0.5696 or 56.96%.
1350
0.4304 × (1 – 0.4304)
The standard error is st.err. ~
- 0.0135 or
1350
1.35%.
The confidence interval for Candidate A goes from 43.03 – 2 x 1.35 to
43.03 + 2 x 1.35, that is from 40.33% to 45.73%.
The confidence interval for Candidate B goes from 56.96 – 2 × 1.35 to
56.96 + 2 x 1.35, that is from 54.26% to 59.66%.
As we can see below, the confidence interval for Candidate B is com-
pletely to the right of the interval for Candidate A. Candidate B now has a
statistically significant advantage over Candidate A.
A
士
40.33
43.03
45.73
В
E*******I*******I
54.26
56.96
59.66
Figure 4.19: Non-overlapping Confidence Intervals for Two Candidates
Both candidates still have one more week to go before the election. Who
knows what may happen.
Transcribed Image Text:Example 4.4.7. One week before the election a new public poll is taken. In the two weeks between the polls, Candidate B increases his support among likely voters. The results of the new poll are 581 in favor of Candidate A, and 769 in favor of Candidate B. Is the difference now statistically significant? CHAPTER 4. STATISTICS 154 The sample size is n = 1350, and the poll presents only two options, so there is a common standard error. 581 For Candidate A, we have k = 581, so 2 0.4303 or 43.03%. For 1350 769 Candidate B, we have k = 769, so p = 2 0.5696 or 56.96%. 1350 0.4304 × (1 – 0.4304) The standard error is st.err. ~ - 0.0135 or 1350 1.35%. The confidence interval for Candidate A goes from 43.03 – 2 x 1.35 to 43.03 + 2 x 1.35, that is from 40.33% to 45.73%. The confidence interval for Candidate B goes from 56.96 – 2 × 1.35 to 56.96 + 2 x 1.35, that is from 54.26% to 59.66%. As we can see below, the confidence interval for Candidate B is com- pletely to the right of the interval for Candidate A. Candidate B now has a statistically significant advantage over Candidate A. A 士 40.33 43.03 45.73 В E*******I*******I 54.26 56.96 59.66 Figure 4.19: Non-overlapping Confidence Intervals for Two Candidates Both candidates still have one more week to go before the election. Who knows what may happen.
42. Show that in Example 4.4.7, page 153, the difference between the two
candidates is statistically significant even at a confidence level of 99.7%.
Transcribed Image Text:42. Show that in Example 4.4.7, page 153, the difference between the two candidates is statistically significant even at a confidence level of 99.7%.
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