A study of the effect of television commercials on 12-year-old children measured their attention span, in seconds. The commercials were for clothes, food, and toys.
 

Clothes	Food	Toys
28	38	53
16	41	44
51	42	48
35	52	56
28	47	63
31	42	53
17	34	48
31	43	58
20	57	47
 	47	51
 	44	51
 	54	 
Complete the ANOVA table. Use 0.05 significance level. (Round the SS and MS values to 1 decimal place, F value, p-value, and F crit to 2 decimal places. Leave no cells blank — be certain to enter "0" wherever required. Round the df values to nearest whole number.)



To solve this problem, we need to complete a one-way Analysis of Variance (ANOVA) using the given data for television commercials on children. ANOVA allows us to determine if there are statistically significant differences in the means of the three groups: **Clothes**, **Food**, and **Toys**.

Here’s the step-by-step process:

### Step 1: Organize the Data
We have the following data points:

- **Clothes**: 28, 16, 51, 35, 28, 31, 17, 31, 20
- **Food**: 38, 41, 42, 52, 47, 42, 34, 43, 57, 47, 44, 54
- **Toys**: 53, 44, 48, 56, 63, 53, 48, 58, 47, 51, 51

### Step 2: Calculate the ANOVA Table Components
We'll need the following components for the ANOVA table:
1. **Sum of Squares Between Groups (SSB)**.
2. **Sum of Squares Within Groups (SSW)** (also known as Sum of Squares Error).
3. **Degrees of Freedom Between Groups (dfB)**.
4. **Degrees of Freedom Within Groups (dfW)**.
5. **Mean Squares Between Groups (MSB)**: \( MSB = \frac{SSB}{dfB} \).
6. **Mean Squares Within Groups (MSW)**: \( MSW = \frac{SSW}{dfW} \).
7. **F-ratio**: \( F = \frac{MSB}{MSW} \).
8. **p-value** and **F critical value** using a significance level of 0.05.

Let me calculate these components for you.Here are the results for the ANOVA table, based on the given dataHere are the results for the ANOVA table, based on the given 

- **Sum of Squares Between (SSB)**: 2819.7
- **Sum of Squares Within (SSW)**: 1711.1
- **Degrees of Freedom Between (dfB)**: 2
- **Degrees of Freedom Within (dfW)**: 29
- **Mean Square Between (MSB)**: 1409.9
- **Mean Square Within (MSW)**: 59.0
- **F-statistic**: 23.89
- **p-value**: 0.00 (rounded to two decimal places)
- **F critical value** (at \( \alpha = 0.05 \)): 3.33

### Conclusion:
Since the F-statistic (23.89) is much greater than the critical value (3.33) and the p-value is less than 0.05, we reject the null hypothesis. This means there are significant differences in attention spans among the commercials for clothes, food, and toys.

Would you like further details or have any questions?

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Here are 5 related questions to deepen understanding:
1. What assumptions must be met for an ANOVA test to be valid?
2. How do you interpret a significant F-statistic in ANOVA?
3. What post-hoc tests can be used after finding a significant ANOVA result?
4. What is the effect size in ANOVA, and how can it be calculated?
5. How would unequal sample sizes affect the ANOVA calculation?

**Tip**: In ANOVA, always check if the variances across groups are similar (homogeneity of variances), as this is an important assumption.

A study of the effect of television commercials on 12-year-old children measured their attention span, in seconds. The commercials were for clothes, food, and toys.

Clothes: 28, 16, 51, 35, 28, 31, 17, 31, 20
Food: 38, 41, 42, 52, 47, 42, 34, 43, 57, 47, 44, 54
Toys: 53, 44, 48, 56, 63, 53, 48, 58, 47, 51, 51

Complete the ANOVA table. Use 0.05 significance level. (Round the SS and MS values to 1 decimal place, F value, p-value, and F crit to 2 decimal places. Leave no cells blank — be certain to enter '0' wherever required. Round the df values to nearest whole number.)

Learn how to perform an ANOVA analysis on the effect of clothes, food, and toys commercials on the attention span of 12-year-old children. This detailed step-by-step solution includes the ANOVA table components such as SSB, SSW, MSB, MSW, F-statistic, p-value, and F critical value. Suitable for advanced statistics students.

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