Question

a survey was conducted to find if modern young people want to have many children. 457 young men were asked if they want to have more than one child. 45% of them answered \yes\.
calculate the margin of error for this survey. round the percentage to one decimal place. do not round intermediate steps.
if 45% of 457 young men want to have more than one child, the margin of error is %.

Explanation:

Step1: Identify the formula for margin of error

The formula for margin of error (ME) for a proportion is \(ME = z\sqrt{\frac{p(1 - p)}{n}}\). For a 95% confidence level (common in surveys when not specified otherwise), \(z = 1.96\), \(p=0.45\) (the proportion of "Yes" responses), and \(n = 457\) (sample size).

Step2: Calculate \(p(1 - p)\)

$$
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$$

Step3: Calculate \(\frac{p(1 - p)}{n}\)

$$ \frac{0.2475}{457}\approx0.000541575 $$

Step4: Calculate \(\sqrt{\frac{p(1 - p)}{n}}\)

$$ \sqrt{0.000541575}\approx0.02327 $$

Step5: Calculate the margin of error

$$ ME=1.96\times0.02327\approx0.0456 $$

To convert to a percentage, multiply by 100: \(0.0456\times100 = 4.6\%\) (rounded to one decimal place)

Answer:

\(4.6\)