Problem 6 - An Introduction to Psychological Statistics

Question

You know that stores tend to charge different prices for similar or identical products, and you want to test whether or not these differences are, on average, statistically significantly different. You go online and collect data from 3 different stores, gathering information on 15 products at each store. You find that the average prices at each store are: Store 1 xbar = $$\$ 27.82$$, Store 2 xbar \(=\$ 38.96\), and Store 3 xbar \(=\$ 24.53\). Based on the overall variability in the products and the variability within each store, you find the following values for the Sums of Squares: \(\mathrm{SST}=683.22, \mathrm{SSW}=441.19 .\) Complete the ANOVA table and use the 4 step hypothesis testing procedure to see if there are systematic price differences between the stores.

Step-by-Step Solution

Answer

The average prices at the stores are statistically significantly different.

Step 1: Set Up Hypotheses

We begin by setting up our null and alternative hypotheses for an ANOVA test. The null hypothesis \((H_0)\) states that the average prices across all stores are equal: \(\mu_1 = \mu_2 = \mu_3\). The alternative hypothesis \((H_a)\) suggests that at least one store has a different average price: At least one of \(\mu_1, \mu_2, \mu_3\) is different.

Step 2: Find Degrees of Freedom

Calculate the degrees of freedom for the numerator \((df_{between})\) and the denominator \((df_{within})\). We have three groups (stores) and a total sample size of 45 products. Thus, \(df_{between} = k - 1 = 3 - 1 = 2\) and \(df_{within} = n - k = 45 - 3 = 42\), where \(k\) is the number of groups and \(n\) is the total sample size.

Step 3: Calculate Mean Squares

Next, calculate the Mean Square Between \((MSB)\) and Mean Square Within \((MSW)\). \(MSB = \frac{SST}{df_{between}} = \frac{683.22}{2} = 341.61\). \(MSW = \frac{SSW}{df_{within}} = \frac{441.19}{42} \approx 10.50\).

Step 4: Compute F-statistic

Calculate the F-statistic using \(F = \frac{MSB}{MSW} = \frac{341.61}{10.50} \approx 32.53\). This F-value is used to determine significance.

Step 5: Determine Critical Value and Decision

Find the critical F-value from the F-distribution table using \(df_{between} = 2\) and \(df_{within} = 42\) at \(\alpha = 0.05\). Suppose the critical value is around 3.22. Since our calculated F-statistic \(32.53\) is greater than the critical value \(3.22\), we reject the null hypothesis.

Step 6: Conclusion

Since the F-statistic \(32.53\) exceeds the critical value, there is statistically significant evidence at \(\alpha = 0.05\) to conclude that the average prices between stores are not equal.

Key Concepts

Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to make decisions or inferences about population parameters based on sample data. In hypothesis testing, we begin by stating two hypotheses:

• The **null hypothesis** \(H_0\): This is the default assumption that there is no effect or difference. For our pricing example, \(H_0\) states that the average price at all three stores is the same.
• The **alternative hypothesis** \(H_a\): This is the hypothesis that there is an effect or a difference. Here, \(H_a\) suggests that at least one of the store prices differs.

The goal is to determine whether there is enough statistical evidence to reject the null hypothesis in favor of the alternative hypothesis. This is done through a series of steps involving calculations, such as computing the test statistic and comparing it to a critical value from a statistical distribution relevant to our test.

Degrees of Freedom

Degrees of freedom (df) is an important concept in hypothesis testing and ANOVA, which represents the number of independent values or quantities that can vary in the calculation of a statistic. It's crucial for determining the appropriate statistical test to use and its critical values.

• **Degrees of Freedom Between Groups (df_between):** This is the number of groups minus one. In our exercise, with 3 stores, it is calculated as \(df_{between} = k - 1 = 3 - 1 = 2\).
• **Degrees of Freedom Within Groups (df_within):** It is the total number of observations minus the number of groups, expressed as \(df_{within} = n - k = 45 - 3 = 42\), where \(n\) is the total sample size.

The greater the degrees of freedom, the more reliable the statistical test results tend to be, as they represent larger or more varied data sets.

F-statistic

The F-statistic is a ratio that compares the variance between group means to the variance within the groups. This ratio is used to determine if the means across different groups are significantly different.

The formula for calculating the F-statistic in ANOVA is:\[ F = \frac{MSB}{MSW} \] Where:

• **MSB (Mean Square Between):** This represents the variance between the groups, which is calculated by dividing the sum of squares between (\(SST\)) by the degrees of freedom between (\(df_{between}\)).
• **MSW (Mean Square Within):** This represents the variance within the groups, calculated by dividing the sum of squares within (\(SSW\)) by the degrees of freedom within (\(df_{within}\)).

A higher F-statistic value suggests a greater disparity between group means, indicating a higher likelihood that at least one group mean is statistically different from the others.

Null and Alternative Hypothesis

The null and alternative hypotheses are central to hypothesis testing and provide a framework for analysis and decision-making.

• **Null Hypothesis (H0):** In the context of the ANOVA test for our exercise, the null hypothesis states that the means of all groups (or stores in this scenario) are equal, suggesting no significant difference in prices.
• **Alternative Hypothesis (Ha):** The alternative hypothesis counters the null, suggesting that at least one group's mean is different, pointing towards a variation in prices between the stores.

Choosing the right hypothesis is critical as it guides the direction of the statistical test and provides a basis for rejecting or not rejecting the null hypothesis. The test aims to use sample data to infer whether any observed differences are enough to reject the null in favor of the alternative hypothesis, often by computing test statistics like the F-statistic and comparing it against critical values for decision-making.