Final answer:
The 95% confidence interval for the mean housing price of 41 homes is calculated using the formula for the sample mean plus or minus the z-score times the standard deviation divided by the square root of the sample size. The interval of $500,000 ± $7653.61 suggests the true mean is between $492,346.39 and $507,653.61 with 95% confidence.
Step-by-step explanation:
Constructing a 95% Confidence Interval for Mean Housing Prices
To construct a 95% confidence interval for the mean housing price in a New York neighborhood, we consider a random sample of 41 homes, with a mean price of $500,000 and a standard deviation of $25,000. As the sample size is relatively large (n>30), we can use the central limit theorem which allows us to assume that the sampling distribution of the sample mean is approximately normal.
For a 95% confidence interval, we find the critical Z-score corresponding to the tails of the standard normal distribution. Since we want a 95% confidence interval, we look up the critical value for 97.5% (to account for the two tails) which is approximately 1.96.
The formula for the confidence interval is:
Confidence Interval = Sample Mean ± (Z-score * (Standard Deviation / sqrt(n)))
Plugging in the values we have:
Confidence Interval = $500,000 ± (1.96 * ($25,000 / sqrt(41)))
Calculating the margin of error:
Margin of error = 1.96 * ($25,000 / sqrt(41))
= 1.96 * ($25,000 / 6.40312)
= 1.96 * 3904.88
= $7653.61
Therefore, the 95% confidence interval for the mean housing price is:
$500,000 ± $7653.61
This means we are 95% confident that the true mean price of the houses in this neighborhood lies between $492,346.39 and $507,653.61.