Math Problem Statement
Several magazines were grouped into three categories based on what level of education of their readers the magazines are geared towards: high, medium, or low level. Then random samples of the magazines were selected to determine the number of three-plus-syllable words were in the advertising copy, and the data is in the table. Is there enough evidence to show that the mean number of three-plus-syllable words in advertising copy is different for at least one of the education levels? Test at the 5% level.
Number of Three Plus Syllable Words in Advertising Copy
High Education Medium Education Low Education 30 20 7 22 18 9 32 26 9 21 22 9 19 23 7 15 0 8 6 12 9 32 20 9 26 20 8 23 21 8 25 0 8 34 29 9 36 4 8 36 8 9 34 11 8 14 18 9 39 20 7 20 7 9 State the hypotheses. H0: Select an answer
Ha: Select an answer
Calculate the F test statistic. Round to two decimal places. F =
Calculate the p-value. Round to four decimal places. p-value =
State your decision. Since p-value ? 0.05, Select an answer H0.
Interpret the results. At the 5% significance level, there Select an answer evidence to show that Select an answer .
Solution
To test whether there is a difference in the mean number of three-plus-syllable words in advertising copy for at least one education level, we perform a One-Way ANOVA test.
1. Hypotheses
- Null hypothesis (H₀): The mean number of three-plus-syllable words in advertising copy is the same across all education levels.
- Alternative hypothesis (Hₐ): At least one mean is different.
2. Data Table
High | Medium | Low |
---|---|---|
30 | 20 | 7 |
22 | 18 | 9 |
32 | 26 | 9 |
21 | 22 | 9 |
19 | 23 | 7 |
15 | 0 | 8 |
6 | 12 | 9 |
32 | 20 | 9 |
26 | 20 | 8 |
23 | 21 | 8 |
25 | 0 | 8 |
34 | 29 | 9 |
36 | 4 | 8 |
36 | 8 | 9 |
34 | 11 | 8 |
14 | 18 | 9 |
39 | 20 | 7 |
20 | 7 | 9 |
3. Calculate the F-test statistic
We compute the test statistic using the formula for ANOVA: I'll calculate the required values. pandas as pd import scipy.stats as stats
Data
data = { "High": [30, 22, 32, 21, 19, 15, 6, 32, 26, 23, 25, 34, 36, 36, 34, 14, 39, 20], "Medium": [20, 18, 26, 22, 23, 0, 12, 20, 20, 21, 0, 29, 4, 8, 11, 18, 20, 7], "Low": [7, 9, 9, 9, 7, 8, 9, 9, 8, 8, 8, 9, 9, 9, 8, 9, 7, 9] }
Create DataFrame
df = pd.DataFrame(data)
Perform ANOVA
f_statistic, p_value = stats.f_oneway(df["High"], df["Medium"], df["Low"])
Display results
f_statistic, p_value### 4. Results
- F-test statistic (F): 26.14 (rounded to two decimal places)
- p-value: (rounded to four decimal places, 0.0000)
5. Decision
Since the p-value () is less than 0.05, we reject the null hypothesis (H₀).
6. Interpretation
At the 5% significance level, there is strong evidence to show that the mean number of three-plus-syllable words in advertising copy is different for at least one education level.
Would you like me to provide a detailed breakdown of calculations, or conduct further post hoc tests to determine which groups differ?
Here are 5 follow-up questions and one tip:
- What does rejecting the null hypothesis mean in practical terms for advertisers?
- How can we calculate confidence intervals for the group means?
- Should a post hoc test (e.g., Tukey's HSD) be conducted to identify the differences?
- What are the assumptions of ANOVA, and were they satisfied?
- How does sample size affect the results of ANOVA?
Tip: Always verify the assumptions of normality and equal variance when performing ANOVA.
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Math Problem Analysis
Mathematical Concepts
Statistics
Analysis of Variance (ANOVA)
Formulas
F = Between-group variability / Within-group variability
p-value calculation for ANOVA
Theorems
F-distribution in ANOVA
Suitable Grade Level
Undergraduate Level or Advanced High School Statistics
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