25.
p-hat= 417/2306 = 0.181
np(1-p) = 2306(0.181)(0.819) > 10
0.181 +/- 1.645 (0.008) = 0.168 and 0.194
I am 90% confident that the population proportion of adult americans older than 18 years old who have donated in the last 2 years is between 0.168 and 0.194.
26.
p-hat= 496/1153 = 0.43
1153(0.43)(0.57) > 10
0.43 +/- 1.96(0.015) = 0.40 and 0.46
I am 95% confident that the population proportion of retirees in the US 25+ years old is between 0.40 and 0.46.
27.
p-hat = 0.52
1003(0.52)(0.48) > 10
0.52 +/- 1.96(0.016) = 0.489 and 0.551
It’s possible that the proportion is greater than 60% because the confidence interval could possibly not account for all proportions, but it is unlikely.
0.48 +/- 1.96(0.016) = 0.449 and 0.511
28.
p-hat = 0.75
1024(0.75)(0.25) > 10
0.75 +/- 2.58(0.014) = 0.714 and 0.786
It is possible for the proportion to fall below 70% because the confidence interval may not account for all proportions, it is likely.
0.25 +/- 2.58(0.014) = 0.214 and 0.286
29.
0.11 +/- 1.96(0.02) = 0.071 and 0.149
0.11 +/- 2.58(0.02) = 0.058 and 0.162
Increasing the confidence interval increases the margin of error.