Question

The U.S. Census Bureau would like to estimate the average square footage of a new single-family home with a 95% confidence interval and a margin of error within plus or minus 100 square feet. Assuming the standard deviation for the square footage of new single-family homes is 850 square feet, the required sample size is ________.

Answer 1

The sample size is
`n =278`

Explanation:

From the question we are told that

The margin of error is
`E = 100`

The standard deviation is
`\sigma = 850`

Given that the confidence level is 95% then the level of significance is mathematically represented as

`\alpha = (100-95)\%`

`\alpha =0.05`

Next we obtain the critical value
`(\alpha )/(2)` from the normal distribution table

The value is
`Z_{(\alpha )/(2) } = 1.96`

Generally the sample size is mathematically represented as

`n = \frac{(Z_{(\alpha )/(2)})^2}{E^2} * \sigma^2`

=>
`n = ((1.96)^2)/(100^2) * 850^2`

=>
`n =278`