Problem 19 - Basic Statistics for Business and Economics with Student

Question

In a particular television market there are three commercial stations, each with its own evening news program from 6: 00 to 6: 30 p.m. According to a report in this morning's local newspaper, a random sample of 150 viewers last night revealed 53 watched the news on WNAE (channel 5), 64 watched on WRRN (channel 11 ), and 33 on WSPD (channel 13). At the .05 significance level, is there a difference in the proportion of viewers watching the three channels?

Step-by-Step Solution

Answer

There is a significant difference in the proportions of viewers watching the three channels.

Step 1: Define Hypotheses

We start by defining the null and alternative hypotheses. The null hypothesis \( H_0 \) states that there is no difference in the proportion of viewers watching the three channels. The alternative hypothesis \( H_a \) states that there is a difference in the proportion of viewers watching the three channels: \( H_0: p_1 = p_2 = p_3 \), \( H_a: \text{At least one } p_i \text{ is different} \).

Step 2: Identify the Sample Proportions

Calculate the sample proportions for each channel. For WNAE: \( \hat{p}_1 = \frac{53}{150} \), for WRRN: \( \hat{p}_2 = \frac{64}{150} \), and for WSPD: \( \hat{p}_3 = \frac{33}{150} \).

Step 3: Calculate the Expected Frequencies

Under the assumption of the null hypothesis (no difference), calculate the expected frequency for each channel. The total viewers are 150, and with an equal proportion for 3 channels, each expected frequency is 50: \( E_1 = E_2 = E_3 = \frac{150}{3} = 50 \).

Step 4: Use Chi-Square Test Formula

Calculate the chi-square statistic using the formula: \( \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \), where \( O_i \) is the observed frequency and \( E_i \) is the expected frequency. Calculate for each channel and sum them up: \( \chi^2 = \frac{(53-50)^2}{50} + \frac{(64-50)^2}{50} + \frac{(33-50)^2}{50} \).

Step 5: Calculate and Compare to Critical Value

Calculate the chi-square statistic. Compare it to the critical value from the chi-square distribution table with \( 2 \) degrees of freedom at \( 0.05 \) significance level. The critical value is approximately \( 5.991 \).

Step 6: Make a Decision

If the calculated chi-square statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Compute \( \chi^2 = 7.72 \), which is greater than \( 5.991 \). Therefore, we reject the null hypothesis.

Key Concepts

Understanding Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics. It's used to determine if there is enough evidence to reject a claim or hypothesis about a population. Imagine you have a hunch that more people prefer one brand of chocolate over another. Hypothesis testing helps you formalize this curiosity into a testable statement.

Here's how you get started:

• Null Hypothesis (\( H_0 \)): This proposes that there is no effect or difference, like stating there's no preference among viewers for the news channels.
• Alternative Hypothesis (\( H_a \)): This suggests there is an effect or a difference, like one channel being more preferred.

You then collect data and use statistical tests to see if there's enough evidence to support the alternative hypothesis over the null. If data strongly points to a difference that isn't due to random chance, you might reject the null hypothesis.

Basics of Proportion Analysis

Proportion analysis is a handy tool when you need to compare parts of a whole. For example, if you're splitting time spent on different apps during the day, proportion analysis could help you understand how your time is divided.

In the context of our exercise, the proportion of viewers for each channel gives insights into viewer preferences. Here's a breakdown:

• Calculate Proportions: Use the formula \( \hat{p}_i = \frac{X_i}{n} \). Here, \( X_i \) is the number of viewers for each channel, and \( n \) is the total number of viewers.
• Expectation: If viewers were to be equally spread, each channel should capture a third of the viewers, if there are three channels.

These proportions enable statisticians to examine differences from what was expected, thus uncovering any possible irregularities or preferences in viewing habits.

Critical Value and Its Role

The critical value is like a threshold in hypothesis testing. It's the dividing line that tells us whether an observed result is significant enough to reject the null hypothesis. Think of it as a cutoff point. If your test statistic crosses it, you've got evidence to believe that something interesting (and statistically significant) is happening.

To find the critical value in a Chi-square test:

• Determine Degrees of Freedom: This depends on the number of categories minus one. For three channels, it's \( 3 - 1 = 2 \).
• Significance Level: A commonly used level is \( 0.05 \).
• Use Chi-Square Table: Using degrees of freedom and significance level, the table provides a critical value (e.g., \( 5.991 \) for \( 2 \) degrees of freedom at \( 0.05 \) level).

If the calculated test statistic is greater than the critical value, it suggests the need to reject the null hypothesis, indicating a statistically significant difference.