"In Exercises 1–5, (a) identify the claim and state H0 and Ha,
[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: Identify the claim and hypotheses. The claim is that the distribution of sales prices in at least one region is different from the others. The null hypothesis (H₀) states that the distributions of sales prices are the same across all regions. The alternative hypothesis (Hₐ) states that the distribution of sales prices in at least one region is different.

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

Step 3: Calculate the test statistic. To perform the ANOVA test, calculate the between-group variance (variation due to differences between regions) and the within-group variance (variation within each region). Use the formula for the F-statistic:

F = MS

_betweenMS_within, where MS stands for mean square.

Step 4: Determine the critical value and p-value. Using the degrees of freedom for the numerator (number of groups minus 1) and the denominator (total number of observations minus the number of groups), find the critical value from an F-distribution table. Compare the calculated F-statistic to the critical value or use software to find the p-value.

Step 5: Make a decision. If the p-value is less than the significance level (α = 0.05), reject the null hypothesis (H₀) and conclude that the distribution of sales prices in at least one region is different. Otherwise, fail to reject H₀ 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 two competing hypotheses: the null hypothesis (H0), which states there is no effect or difference, and the alternative hypothesis (Ha), which suggests there is an effect or difference. In this context, the goal is to determine if the sales prices of condominiums differ across regions.

ANOVA (Analysis of Variance)

ANOVA is a statistical technique used to compare the means of three or more groups to see if at least one group mean is different from the others. It helps in determining if the variation in sales prices among different regions is statistically significant. In this case, it would be used to analyze the sales prices of condominiums across the four U.S. regions.

P-value

The p-value is a measure that helps determine the significance of results in hypothesis testing. 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 are different from the others.

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