Question

In a poll, 51% of the people polled answered yes to the question "Are you in favor of the death penalty for a person convicted of murder?" The margin of error in the poll was 2%, and the estimate was made with 95% confidence. At least how many people were surveyed?

Click here to view the standard normal distribution table (page 1.
Click here to view the standard normal distribution table (page 2).

The minimum number of surveyed people was. (Round up to the nearest integer.)

Answer 1

386

Explanation:

To secure a 95% confidence level with a 5% margin of error and ensure the estimated percentage of people in favor of the death penalty is accurately within the specified range, a minimum of 386 people must be surveyed. This determination takes into account the margin of error and confidence level of the poll.

The 5% margin of error allows for a variation of up to 5% from the estimated percentage of people supporting the death penalty, which is set at 51%. The 95% confidence level reflects the desire to be 95% confident that the actual percentage falls within this estimated range.

The calculation for the minimum sample size employs the formula:

`\[ n = \frac{{Z^2 \cdot p \cdot q}}{{E^2}} \]`

Where:

n is the sample size,

Z is the Z-score corresponding to the desired confidence level (approximately 1.96 for 95% confidence),

p is the estimated proportion (51% expressed as 0.51),

q is
`\( 1 - p \)`,

E is the margin of error (5% expressed as 0.05).

Plugging in these values:

`\[ n = \frac{{1.96^2 \cdot 0.51 \cdot 0.49}}{{0.05^2}} \]`

`\[ n = 385.78 \]`

Therefore, the minimum number of people surveyed to meet these criteria is approximately 386.