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	 Compute the 95% confidence intervals that estimate the difference between each pair of means. (Negative amount should be indicated by a minus sign. Round your answers to 2 decimal places.)

To compute the 95% confidence intervals for the differences between each pair of means, we need to follow these steps:

### Step 1: Calculate the means for each category (Clothes, Food, and Toys).

We will compute the mean of the attention span for each group (Clothes, Food, and Toys).

### Step 2: Compute the standard deviations for each group.

We need to calculate the standard deviation for the three groups to assess the variability in the data.

### Step 3: Calculate the standard error for each pair of means.

The standard error of the difference between two means is given by the formula:

\[
SE = \sqrt{\left(\frac{SD_1^2}{n_1}\right) + \left(\frac{SD_2^2}{n_2}\right)}
\]
Where:
- \(SD_1\) and \(SD_2\) are the standard deviations of the two groups.
- \(n_1\) and \(n_2\) are the sample sizes of the two groups.

### Step 4: Compute the margin of error.

The margin of error is computed using the following formula:

\[
ME = t \times SE
\]

Where \(t\) is the critical value from the t-distribution for a 95% confidence level, based on the degrees of freedom. For large samples (which we approximate when \(n \geq 30\)), the \(t\)-value is close to 1.96.

### Step 5: Compute the confidence interval for the difference between each pair of means.

The confidence interval is given by:

\[
\text{Confidence Interval} = (\text{Mean Difference} - \text{ME}, \text{Mean Difference} + \text{ME})
\]

Now, let's compute these values.

I will start by calculating the mean, standard deviation, and confidence intervals for the data you provided.### Summary of Results:

- **Mean values:**
  - Clothes: 30.40 seconds
  - Food: 45.75 seconds
  - Toys: 52.22 seconds

- **95% Confidence Intervals for the differences between means:**
  1. **Clothes vs. Food**: (-24.99, -5.71)
  2. **Clothes vs. Toys**: (-31.54, -12.11)
  3. **Food vs. Toys**: (-12.97, 0.03)

These confidence intervals represent the estimated difference between the means of attention span for the given commercials. A negative value indicates that the first group (e.g., Clothes) has a lower mean than the second group (e.g., Food).

Would you like more details or have any questions?

### Related Questions:
1. How is the t-critical value for confidence intervals determined?
2. What are the effects of standard deviation on the width of confidence intervals?
3. How can you interpret a confidence interval that includes zero?
4. Why are the confidence intervals for "Clothes vs. Toys" wider than "Food vs. Toys"?
5. How does sample size affect the precision of a confidence interval?

### Tip:
Confidence intervals become narrower with larger sample sizes or lower variability, improving the precision of your estimates.

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. Compute the 95% confidence intervals that estimate the difference between each pair of means. (Negative amount should be indicated by a minus sign. Round your answers to 2 decimal places.)

Calculate the 95% confidence intervals for differences in attention span between commercials for clothes, food, and toys for 12-year-old children. This statistical analysis involves key concepts such as means, standard deviation, and the t-distribution. Suitable for advanced high school statistics.

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