Question

6. With the origin at Champaign, the coordinates for Springfield and Bloomington on the map shown are (-74, -21) and

(-29, 27), respectively. Assuming the coordinates are given in miles, find the distance between the two cities. Round to
the nearest tenth of a mile. Show your work!
55
80
70
74
65
65
7. A recent news article reported that according to a poll, only 29% of Americans support a proposed infrastructure plan.
At the bottom of the article was the following sentence: "The Post-ABC poll was conducted by telephone Jan. 12-15,
2017, including landline and cellphone respondents. Overall results have a margin of sampling error of plus or minus
3.5 percentage points." The confidence level was not given. Supposing the confidence level was 95%, write a sentence
or two explaining the results of the poll.
For Questions 8 and 9: In conducting a survey of 200 students, a researcher found that 22% had significant (more than $1,000)
credit card debt.
8. Calculate the margin of error with a 95% confidence level for this survey.
9. If the researcher wanted to get a margin of error of 4%, how many total students would she need to survey? Assume the
number of students with significant credit card debt stays at 22%.

Answer 1

6. To find the distance between Springfield and Bloomington, we can use the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) = (-74, -21) and (x2, y2) = (-29, 27). Substituting these values into the formula, we get:

d = sqrt((-29 - (-74))^2 + (27 - (-21))^2) = sqrt(45^2 + 48^2) = sqrt(4059) ≈ 63.7

Therefore, the distance between Springfield and Bloomington is approximately 63.7 miles.

7. With a confidence level of 95% and a margin of error of plus or minus 3.5 percentage points, we can interpret the results of the poll as follows: if the same poll were conducted many times, 95% of the time the percentage of Americans who support the proposed infrastructure plan would fall within 3.5 percentage points of the reported value of 29%. In other words, the true percentage of Americans who support the plan is likely to be between 25.5% and 32.5%, with a high degree of confidence.

8. To calculate the margin of error with a 95% confidence level, we can use the formula:

margin of error = zsqrt(p(1-p)/n)

where z is the z-score associated with a 95% confidence level (which is approximately 1.96), p is the proportion of students with significant credit card debt (which is 0.22), and n is the sample size (which is 200). Substituting these values into the formula, we get:

margin of error = 1.96sqrt(0.22(1-0.22)/200) ≈ 0.07

Therefore, the margin of error for this survey with a 95% confidence level is approximately 0.07, or 7%.

9. To find the sample size needed to achieve a margin of error of 4%, we can use the formula:

n = (zsqrt(p(1-p))/e)^2

where z is the z-score associated with a 95% confidence level (which is approximately 1.96), p is the proportion of students with significant credit card debt (which is 0.22), and e is the desired margin of error (which is 0.04). Substituting these values into the formula, we get:

n = (1.96sqrt(0.22(1-0.22))/0.04)^2 ≈ 1385

Therefore, the researcher would need to survey approximately 1385 students to achieve a margin of error of 4%.

Explanation: