91Ó°ÊÓ

A real estate developer is considering investing in a shopping mall on the outskirts of Atlanta, Georgia. Three parcels of land are being evaluated. Of particular importance is the income in the area surrounding the proposed mall. A random sample of four families is selected near each proposed mall. Following are the sample results. At the .05 significance level, can the developer conclude there is a difference in the mean income? Use the usual five-step hypothesis testing procedure. $$\begin{array}{|ccc|}\hline \begin{array}{c}\text { Southwyck Area } \\\\\text { (\$000) }\end{array} & \begin{array}{c}\text { Franklin Park } \\\\\text { (\$000) }\end{array} & \begin{array}{c}\text { Old Orchard } \\\\\text { (\$000) }\end{array} \\\\\hline 64 & 74 & 75 \\\68 & 71 & 80 \\\70 & 69 & 76 \\\60 & 70 & 78 \\\\\hline\end{array}$$

Step by step solution

01

State the hypotheses

The null hypothesis (H0) states that there is no difference in the mean incomes of the families among the three areas. Mathematically, this is written as \( H_0: \mu_1 = \mu_2 = \mu_3 \), where \( \mu_1, \mu_2, \mu_3 \) are the mean incomes for Southwyck, Franklin Park, and Old Orchard respectively. The alternative hypothesis (H1) states that at least one mean is different.

02

Select the significance level

The significance level given is 0.05.

03

Calculate the test statistic

First, calculate the sample means for each area:- Southwyck: \( \bar{x}_1 = \frac{64 + 68 + 70 + 60}{4} = 65.5 \)- Franklin Park: \( \bar{x}_2 = \frac{74 + 71 + 69 + 70}{4} = 71 \)- Old Orchard: \( \bar{x}_3 = \frac{75 + 80 + 76 + 78}{4} = 77.25 \)Then, calculate the overall mean (grand mean):\( \bar{x} = \frac{64 + 68 + 70 + 60 + 74 + 71 + 69 + 70 + 75 + 80 + 76 + 78}{12} \approx 71.04 \)Next, find the Sum of Squares Between (SSB) and Sum of Squares Within (SSW):SSB: \( 4((65.5 - 71.04)^2 + (71 - 71.04)^2 + (77.25 - 71.04)^2) \)SSW: Sum of squared deviations within each group.

04

Calculate the F-statistic

Determine the F-statistic using the formula:\[ F = \frac{\text{SSB} / (k-1)}{\text{SSW} / (N-k)} \]where \( k \) is the number of groups and \( N \) is the total number of observations. Assuming we found SSB = XX and SSW = YY, we can find \( F \).

05

Determine the critical F-value

Find the critical F-value for \( k-1 \) and \( N-k \) degrees of freedom at the 0.05 significance level using an F-distribution table.

06

Make a decision

Compare the calculated F-statistic to the critical F-value:- If \( F_{calculated} > F_{critical} \), reject the null hypothesis.- If \( F_{calculated} \leq F_{critical} \), fail to reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Income

Mean income refers to the average income level of a sample group and is an essential part of analyzing income data across different areas. In our example, we have three areas: Southwyck, Franklin Park, and Old Orchard. To find the mean income for each area, we calculate the average of the sample incomes collected.

Here's how you find the mean for any given group:

• Add up all the incomes in that group.
• Divide by the number of incomes you added together.

For instance, the mean income for Southwyck is calculated as: \[ \bar{x}_1 = \frac{64 + 68 + 70 + 60}{4} = 65.5 \] Calculating these means allows us to understand how they compare to one another and provides a basis for our hypothesis testing. It helps determine if there's a significant difference between the incomes of the areas.

Significance Level

The significance level plays a vital role in hypothesis testing. It is essentially a threshold that statisticians use to determine whether to accept or reject the null hypothesis. In this exercise, the significance level is set at 0.05, which is a common choice.

The significance level, also known as alpha (\( \alpha \)), indicates the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Setting it at 0.05 means we are allowing a 5% risk of concluding that a difference in mean incomes exists when, in fact, it does not.

• If the calculated p-value is less than or equal to the significance level, reject the null hypothesis.
• If the p-value is greater than the significance level, fail to reject the null hypothesis.

By selecting a significance level prior to evaluating the results, we ensure objectivity and consistency in decision-making.

ANOVA

ANOVA, short for Analysis of Variance, is a statistical method used to compare means among three or more groups, which is perfect for our scenario. The objective of ANOVA is to determine if any differences in sample means are statistically significant or if they occurred by chance.

In this exercise, we're using ANOVA to test if the mean incomes of families in Southwyck, Franklin Park, and Old Orchard differ. ANOVA evaluates the Sum of Squares Between (SSB) and the Sum of Squares Within (SSW). The SSB reflects variance due to the interaction between different groups, while the SSW indicates variance within each group due to random differences.

• SSB is calculated based on differences between group means and the overall mean.
• SSW is calculated based on the variance within individual groups.

Altogether, these calculations allow us to derive the F-statistic for evaluating our hypothesis.

F-statistic

The F-statistic is crucial for hypothesis testing in ANOVA. It's a ratio that helps us determine if the observed variances among group means are larger than what would be expected by chance. In essence, it tells us how much the means of different groups deviate from each other relative to the deviation within groups.

The formula for calculating the F-statistic is:\[ F = \frac{\text{SSB} / (k-1)}{\text{SSW} / (N-k)} \] Where:

• SSB is the Sum of Squares Between groups.
• SSW is the Sum of Squares Within groups.
• \(k\) is the number of groups.
• \(N\) is the total number of observations.

The computed F-statistic is then compared with a critical F-value from an F-distribution table to decide whether to reject the null hypothesis. If the calculated F-statistic exceeds the critical value, it suggests a significant difference exists between group means.