If n=160 and p̂=0.55, construct a 90% confidence i

Questions: If n=160 and p̂=0.55, construct a 90% confidence interval about the population proportion. Round your answers to three decimal places. Preliminary: a. Is it safe to assume that n ≤ 0.05 of all subjects in the population? No Yes b. Verify n p̂(1-p̂) ≥ 10. Round your answer to one decimal place. n p̂(1-p̂)= Confidence Interval: What is the 90% confidence interval to estimate the population proportion? Round your answer to three decimal places. <p<

Transcript text: If $n=160$ and $\widehat{p}=0.55$, construct a $90 \%$ confidence interval about the population proportion. Round your answers to three decimal places. Preliminary: a. Is it safe to assume that $n \leq 0.05$ of all subjects in the population? No Yes b. Verify $n \widehat{p}(1-\hat{p}) \geq 10$. Round your answer to one decimal place. \[ n \widehat{p}(1-\widehat{p})= \] Confidence Interval: What is the $90 \%$ confidence interval to estimate the population proportion? Round your answer to three decimal places. $

Solution

Solution Steps

Step 1: Preliminary Check for Population Assumption

To determine if it is safe to assume that $ n \leq 0.05 $ of all subjects in the population, we note that we do not have the total population size. Therefore, we can conclude that it is not safe to make this assumption.

Step 2: Verification of $ n \widehat{p}(1-\widehat{p}) $

We calculate $ n \widehat{p}(1-\widehat{p}) $ to verify if it meets the condition $ n \widehat{p}(1-\widehat{p}) \geq 10 $:

\[ n \widehat{p}(1-\widehat{p}) = 160 \times 0.55 \times (1 - 0.55) = 39.6 \]

Since $ 39.6 \geq 10 $, the condition is satisfied.

Step 3: Confidence Interval Calculation

We compute the $ 90\% $ confidence interval for the population proportion $ p $ using the sample proportion $ \widehat{p} = 0.55 $ and sample size $ n = 160 $. The resulting confidence interval is:

\[ 0.485 < p < 0.615 \]

Final Answer

The answers to the questions are as follows:

a. Is it safe to assume that $ n \leq 0.05 $ of all subjects in the population? No
b. Verify $ n \widehat{p}(1-\widehat{p}) \geq 10 $: $ n \widehat{p}(1-\widehat{p}) = 39.6 $
Confidence Interval: $ 0.485 < p < 0.615 $

Thus, the final boxed answers are: \[ \boxed{\text{a: No, b: 39.6, Confidence Interval: } 0.485 < p < 0.615} \]

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