Fair Market Rent. According to the document Out of Reach,...

Fair Market Rent. According to the document Out of Reach, published by the National Low Income Housing Coalition, the fair market rent (FMR) for a two-bedroom unit in Maine is $\$ 779 .$ A sample of 100 randomly selected two-bedroom units in Maine yielded the data on monthly rents, in dollars, given on the WeissStats CD. At the $5 \%$ significance level, do the data provide sufficient evidence to conclude that the mean monthly rent for two-bedroom units in Maine is greater than the FMR of $\$ 779 ?$ Perform the required hypothesis test both with and without the outlier.

Key Concepts

P-Value

The p-value quantifies the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. A low p-value (typically less than the significance level) suggests that the observed data are unlikely under the null hypothesis, leading researchers to reject H0 in favor of the alternative hypothesis.

Impact of Outliers

Outliers are data points that differ significantly from other observations in the dataset. They can substantially influence statistical measures like the mean and standard deviation, potentially distorting the outcome of tests such as the t-test. Analyzing data both with and without outliers helps to assess their effect on the overall results and ensures that conclusions are robust and reliable.

One-Sample t-Test

The one-sample t-test is used when comparing the mean of a sample to a known value or benchmark, especially when the population standard deviation is unknown and the sample size is relatively small. This test calculates a t-statistic based on the sample mean, sample standard deviation, and sample size, and it determines how far the sample mean deviates from the hypothesized mean in units of estimated standard error.

Significance Level

The significance level, often designated as alpha (?), sets the threshold for rejecting the null hypothesis. It indicates the maximum acceptable probability of making a Type I error, which is the error of rejecting a true null hypothesis. In many studies, including this case, an ? of 5% is used, meaning that there is a 5% risk of mistakenly concluding a significant effect when none exists.

Null and Alternative Hypotheses

The null hypothesis (H0) typically represents a status quo or a statement of no difference, while the alternative hypothesis (H1) represents the claim under investigation. In tests comparing a sample mean to a given benchmark, H0 often posits that the sample mean is less than or equal to the benchmark, and H1 asserts that it is greater. This setup is central to making valid statistical decisions.

Hypothesis Testing

Hypothesis testing is a systematic method used in statistics to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves framing a null hypothesis, which represents no effect or difference, and an alternative hypothesis, which reflects the claim being tested. Statistical tests then evaluate the probability of observing the sample data if the null hypothesis were true.

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