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.