Question

In a comparison of the prices of items at five supermarkets, six items were randomly selected and the price of each was recorded for each of the five supermarkets. The objective of the study was to see whether the data indicated differences in the levels of prices among the five supermarkets. The prices are listed in the table. a. Does the distribution of the prices differ from one supermarket to another? Test using the Friedman \(F_{r}\) -test with \(\alpha=.05 .\) b. Find the approximate \(p\) -value for the test and interpret it.

Short answer

A. \(F_{r} = \frac{12}{k(n+1)} \sum_{j=1}^k R_j^2 - \frac{3(k+1)n}{n+1}\) B. \(F_{r} = \frac{12}{k(n-1)} \sum_{j=1}^k R_j^2 - \frac{3(k+1)n}{n-1}\) C. \(F_{r} = \frac{12}{k(n-1)} \sum_{j=1}^k R_j^2 + \frac{3(k+1)n}{n+1}\) D. \(F_{r} = \frac{12}{k(n+1)} \sum_{j=1}^k R_j^2 + \frac{3(k+1)n}{n-1}\)

Step-by-step solution

Gather the data

The first step is to gather the data from the given table. Organize the prices into rows and columns, with each row representing an item and each column a supermarket.

Rank the prices for each item across supermarkets

For each item, rank the prices from the lowest (1) to the highest (5). If there are any ties, use the average rank. Calculate the sum of ranks for each supermarket and the average rank for each item.

Calculate the Friedman test statistic

The Friedman test statistic, \(F_{r}\), is given by the formula: \(F_{r} = \frac{12}{k(n+1)} \sum_{j=1}^k R_j^2 - \frac{3(k+1)n}{n+1}\) where \(k\) is the number of supermarkets, \(n\) is the number of items, and \(R_j\) is the sum of ranks for each supermarket. Calculate \(F_{r}\) using the data from the previous step.

Determine the critical value

Since we want to test at the 5% significance level (\(\alpha = 0.05\)), we need to find the critical value. Use the Chi-squared (\(\chi^2\)) distribution table and find the critical value for \(k-1\) degrees of freedom.

Compare the test statistic and critical value

Compare the calculated test statistic, \(F_{r}\), with the critical value found in the previous step. If the test statistic is higher than the critical value, reject the null hypothesis and conclude that there is a significant difference in price distribution among the five supermarkets. Otherwise, fail to reject the null hypothesis and conclude that there is no significant difference in price distribution.

Find the approximate p-value

The p-value can be found by using a Chi-squared (\(\chi^2\)) distribution calculator or software. Input the test statistic value and the degrees of freedom (k-1) into the calculator to find the p-value.

Interpret the p-value

Interpret the p-value found in the previous step. If the p-value is less than the significance level (\(\alpha = 0.05\)), reject the null hypothesis and conclude that there is a significant difference in price distribution among the five supermarkets. Otherwise, fail to reject the null hypothesis and conclude that there is no significant difference in price distribution.

Key concepts

Non-parametric Test

Non-parametric tests are statistical tests that do not assume a specific distribution for the data. This makes them versatile and applicable to a wide range of data types. For example, they are useful when the data doesn't meet the normal distribution requirements. This flexibility comes in handy for diverse datasets, such as those involving rankings or ordinal data.

In this problem, we are dealing with supermarket prices which may not follow a normal distribution. Hence, applying a non-parametric test, like the Friedman Test, is ideal.

Advantages of non-parametric tests include:

• No need to meet stringent assumptions about data distribution.
• Can be used with small sample sizes.
• Capable of handling outliers effectively.

By using a non-parametric approach, you can rigorously test hypotheses without being constrained by the underlying data distribution.

Price Comparison

Price comparison involves evaluating the costs of similar products across different points of sale, such as supermarkets in this exercise. With prices collected from multiple sources, it's crucial to determine if variations are random or statistically significant.

The aim is to understand whether there's a consistent price difference among the supermarkets. Each price is treated as a data point, with rankings used to simplify the comparison across different outlets.

By using statistical methods, such as the Friedman Test, we can analyze these price points beyond simple observation, providing a quantifiable basis for making claims about price differences. This systematic approach ensures that conclusions aren't drawn purely from anecdotal evidence.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used to decide whether a claim about a dataset is true or false. It starts with a null hypothesis, which represents no effect or no difference, and an alternative hypothesis, indicating the presence of an effect or difference.

In the context of the Friedman Test, the hypotheses could be:

• Null Hypothesis ( H_0 ): There are no differences in median prices among the five supermarkets.
• Alternative Hypothesis ( H_1 ): At least one supermarket differs in its median price.

The testing process involves calculating a test statistic, comparing it against a critical value from a theoretical distribution table (typically Chi-squared in non-parametric tests like Friedman's), and deciding whether to reject or fail to reject the null hypothesis.

Statistical significance is determined by the p-value, which indicates the probability of observing your data, or something more extreme, if the null hypothesis is true. If this p-value is below a predetermined threshold (often 0.05), the null hypothesis is rejected, indicating statistically significant differences.

Chi-squared Distribution

The Chi-squared distribution is a critical concept used in many hypothesis testing scenarios, especially with non-parametric tests like the Friedman Test. It represents how often different sample variances occur within samples assuming the null hypothesis is true.

In this exercise, the Chi-squared distribution helps find the critical value against which the calculated test statistic ( F_r ) is compared. The distribution has different shapes depending on the degrees of freedom, which is simply the number of independent values that can vary in the final calculation.

For the Friedman Test, the degrees of freedom are calculated as the number of groups (supermarkets) minus one. Looking up this value in a Chi-squared table allows determining whether the test statistic significantly deviates from what was expected under the null hypothesis.

• This distribution is critical for understanding whether observed data significantly deviates from what is expected under the null hypothesis in non-parametric tests.
• The area under the curve beyond the calculated test statistic indicates the p-value, guiding decisions about the statistical significance.

Understanding how the Chi-squared distribution works ensures accurate hypothesis testing and interpretation of results.