Question

The Pew Research Center recently conducted a survey of 1007 U.S. adults and found that 85% of those surveyed know what Twitter is.

Using the survey results construct a 95% confidence interval estimate of the percentage of all adults who know what Twitter is. Round the percentages to the first decimal place.

(c) Explain why it would or would not be okay for a newspaper to make this statement: “Based on results from a recent survey, more than 3 out of 4 U.S. adults know what Twitter is.”

(d) Assuming there had never been a prior survey about U.S. adults knowing what Twitter is, what is the minimum sample size necessary to make a 95% confidence interval estimate of the percentage of adults who know about Twitter? Use the margin of error from part (a).

(e) A common criticism of surveys is that they poll only a very small percentage of the population and therefore cannot be accurate. Based upon your answer to part (d), is a sample of only 1007 adults a sample size that is too small? Write a brief explanation of why the sample size of 1007 is or is not too small.

Answer 1

At 95% confidence interval the critical value is z_0.025 = 1.96

The 95% confidence interval for population proportion is

+/- z_0.025 * sqrt((1 - )/n)

= 0.85 +/- 1.96 * sqrt(0.85 * 0.15/1007)

= 0.85 +/- 0.0221

= 0.8279, 0.8721

= 0.828, 0.872

= 82.8%, 87.2%

c) 3/4 = 0.75 = 75%

Since 75% does not lie in the confidence interval, so it would not be ok for a news paper to make the statement that more than 3 out of 4 U.S adults known what Twitter is.

d) Margin of error = 0.0221

or, z_0.025 * sqrt(p(1 - p)/n) = 0.0221

or, 1.96 * sqrt(0.5 * 0.5/n) = 0.0221

or, n = (1.96 * sqrt(0.5 * 0.5)/0.0221)^2

or, n = 1967

e) Yes, a sample of only 1007 adults a sample size that is too small.