Alright, let's work through this hypothesis testing problem step-by-step.
### Step 1: State the Hypotheses
We need to test whether the average number of pages in the four different types of magazines is the same.
- Null Hypothesis ([tex]\(H_0\)[/tex]): The four magazine types have the same average number of pages.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): At least one magazine type has a different average number of pages.
### Step 2: Gather the Data
The data given for the number of pages in the different magazine types:
- Home decorating: 173, 286, 163, 208, 200
- News: 89, 94, 123, 109, 101
- Health: 83, 156, 90, 108, 96
- Computer: 103, 137
### Step 3: Select the Appropriate Statistical Test
Here, we utilize a one-way ANOVA (Analysis of Variance) test because:
- We are comparing the means of more than two groups.
- The data appears to be measured on an interval or ratio scale.
### Step 4: Calculate the ANOVA Test Statistic and P-value
From the analysis:
- The F-statistic is approximately 10.70
- The P-value is approximately 0.000811
### Step 5: Compare the P-value to the Significance Level
We use a significance level ([tex]\( \alpha \)[/tex]) of 5%, or 0.05.
### Step 6: Draw a Conclusion
Since the P-value (0.000811) is less than the significance level (0.05), we reject the null hypothesis.
### Step 7: Interpret the Results
The results of the ANOVA test suggest that there is a statistically significant difference in the average number of pages among at least one of the four different types of magazines.
Thus, we conclude that at least one type of magazine (home decorating, news, health, or computer) has a different average length of pages than the others.
