A realtor needs to know the average price of a house in a...

A realtor needs to know the average price of a house in a metropolitan area. A random sample of 50 house prices in the metropolitan area yielded $$\bar{x}=\$ 120,000$$ and $$s=\$ 47,000$$. Give a $95 \%$ confidence interval for the average price of a house in the metropolitan area.

Key Concepts

Sampling Distribution of the Mean

The sampling distribution of the mean is the probability distribution of all possible sample means taken from a population. Thanks to the Central Limit Theorem, this distribution approximates a normal distribution when the sample size is sufficiently large, which justifies using normal-based methods or the t-distribution for constructing confidence intervals.

t-Distribution

The t-distribution is a probability distribution used when estimating population parameters when the sample size is relatively small and/or the population standard deviation is unknown. It accounts for additional uncertainty in these conditions by having heavier tails compared to the normal distribution.

Confidence Interval

A confidence interval is a range of estimated values derived from sample data that is likely to include the true population parameter. It represents both the estimate and the uncertainty about that estimate, providing a level of confidence (such as 95%) that the interval contains the population mean.

Margin of Error

The margin of error is the amount added and subtracted from the sample estimate to create the confidence interval. It reflects the variability inherent in the sample and is calculated using the critical value from a probability distribution (like the t-distribution) and the standard error of the sample mean.

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