In Exercises 1–5, (f) interpret the decision in the context of the original claim.
[APPLET] The table shows the sales prices for a random sample of apartment condominiums and cooperatives in four U.S. regions. At , can you conclude that the distribution of the sales prices in at least one region is different from the others? (Adapted from National Association of Realtors)

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Step 1: State the null and alternative hypotheses. The null hypothesis (H₀) is that the distribution of sales prices is the same across all regions. The alternative hypothesis (H₁) is that the distribution of sales prices in at least one region is different from the others.

Step 2: Choose the appropriate statistical test. Since we are comparing the distributions of sales prices across multiple groups (regions), a one-way ANOVA test is appropriate to determine if there are significant differences among the means of the groups.

Step 3: Calculate the test statistic. First, compute the mean sales price for each region and the overall mean. Then, calculate the between-group variability (sum of squares between, SSₐ) and the within-group variability (sum of squares within, SSₑ). Use these to compute the F-statistic using the formula:

F = (SSₐ / dfₐ) / (SSₑ / dfₑ)

, where dfₐ and dfₑ are the degrees of freedom for between-group and within-group variability, respectively.

Step 4: Determine the critical value or p-value. Using the F-distribution table, find the critical value for the given significance level (α = 0.05) and degrees of freedom. Alternatively, calculate the p-value corresponding to the computed F-statistic.

Step 5: Make a decision and interpret the result. If the F-statistic exceeds the critical value or if the p-value is less than α, reject the null hypothesis. Conclude that there is sufficient evidence to suggest that the distribution of sales prices in at least one region is different from the others. Otherwise, fail to reject the null hypothesis and conclude that there is no significant difference in the distributions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (no effect or difference) and an alternative hypothesis (there is an effect or difference). In this context, the null hypothesis would state that the sales prices across the four regions are the same, while the alternative would suggest at least one region has a different distribution of sales prices.

ANOVA (Analysis of Variance)

ANOVA is a statistical technique used to compare the means of three or more groups to determine if at least one group mean is significantly different from the others. In this case, ANOVA would be appropriate to analyze the sales prices of condominiums and cooperatives across the four U.S. regions, helping to identify if regional differences exist in sales prices.

P-value

The p-value is a measure that helps determine the significance of the results obtained from a statistical test. It represents the probability of observing the data, or something more extreme, assuming the null hypothesis is true. A low p-value (typically less than 0.05) indicates strong evidence against the null hypothesis, suggesting that the sales prices in at least one region differ significantly from the others.

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