Solutions

Please note that the steps show rounded numbers, but that the final answers to the problems are calculated without rounding.

Problem Part Solution
1 - \(n \hat{p} \geq 10\)
\(n(1 - \hat{p}) \geq 10\)
\(100(0.12) = 12 \geq 10\)
\(100(1 - 0.12) = 88 \geq 10\)
The requirements are met.
2 - \(\hat{p} = 0.12\)
3 - (0.067, 0.173) We are 90% confident that the true proportion of peanuts in the can is between 6.7% and 17.3%.
4 - \(n p \geq 10\)
\(n(1 - p) \geq 10\)
\(100(0.2) = 20 \geq 10\)
\(100(1 - 0.2) = 80 \geq 10\)
Since both conditions are true, we conclude that \(n\) is sufficiently large so that \(\hat{p}\) will be approximately distributed.
5 - \(\hat{p} = 0.12\)
6 - \(H_0: p = 0.2\)
\(H_a: p < 0.2\)
7 - \(z = -2\)
8 - \(P-value = 0.0228\)
9 - NormalProbApplet
10 - reject the null hypothesis
11 - There is sufficient to suggest that the proportion of peanuts in the can is less than 20%.
12 - (0.023, 0.024) We are 95% confident that the true proportion of the population who die after contracting H1N1 is between 2.3% and 2.4%.
13 - (0.547, 0.639) We are 95% confident that the true proportion of the population who die after contracting H5N1 is between 54.7% and 63.9%.
14 - The bird flu (H5N1) is by far more deadly! More than half of those who contract the bird flu will die, compared to only 2 to 3% of those who are infected with the swine flu. Fortunately for us, the bird flu is currently onlypassed to humans through contact with infected birds. Epidemiologists are concerned about a global pandemic of this disease, which would almost surely happen if the virus mutates to allow human-to-human transmission.
15 - \(n = 423 \text{ people}\)
16 - \(n = 334 \text{ people}\)
17 - pie bar
18 - \(n p \geq 10\)
\(n(1 - p) \geq 10\)
\(800(0.43) = 344 \geq 10\)
\(800(1 - 0.43) = 456 \geq 10\)
Since both conditions are true, we conclude that \(n\) is sufficiently large so that \(\hat{p}\) will be approximately distributed.
19 - \(\hat{p} = 0.405\)
20 - \(H_0: p = 0.43\)
\(H_a: p \neq 0.43\)
21 - \(z = -1.428\)
22 - \(P-value = 0.1532\)
23 - fail to reject the null hypothesis
24 - There is insufficient to suggest that the proportion of adults who received a phishing email in 2012 is different than 43%.