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To find a 98% confidence interval for the difference between the mean speeds of Processor A and Processor B, we’ll follow these steps:

1. **Calculate the differences \( d_i \) for each pair**: Subtract each speed of Processor B from Processor A for each code.
2. **Compute the mean (\( \bar{d} \)) and standard deviation (\( s_d \)) of the differences**.
3. **Determine the critical t-value** for a 98% confidence level with \( n-1 = 5 \) degrees of freedom.
4. **Calculate the margin of error** using \( t \times \frac{s_d}{\sqrt{n}} \).
5. **Construct the confidence interval** by adding and subtracting the margin of error from the mean difference.

Let's go through the calculations.

### Step-by-Step Calculations

#### 1. Differences \( d_i \)
For each code:
- \( d_1 = 22.1 - 25.8 = -3.7 \)
- \( d_2 = 18.5 - 15.3 = 3.2 \)
- \( d_3 = 23.3 - 21.7 = 1.6 \)
- \( d_4 = 16.0 - 23.5 = -7.5 \)
- \( d_5 = 28.3 - 24.7 = 3.6 \)
- \( d_6 = 22.2 - 21.4 = 0.8 \)

#### 2. Mean and Standard Deviation of \( d \)
- Mean \( \bar{d} = \frac{-3.7 + 3.2 + 1.6 - 7.5 + 3.6 + 0.8}{6} \)
- Standard deviation \( s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}} \)

#### 3. Critical t-value
For a 98% confidence interval and 5 degrees of freedom, the critical t-value \( t \) can be found using a t-distribution table or calculator.

#### 4. Margin of Error
Margin of error \( = t \times \frac{s_d}{\sqrt{n}} \)

#### 5. Confidence Interval
The confidence interval for the difference between mean speeds is given by:
\[ \bar{d} \pm \text{Margin of Error} \]

Would you like me to carry out these specific calculations, or do you have any specific parts you’d like more details on?

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**Questions for further exploration:**
1. Would you like to understand how to use the t-table for confidence intervals?
2. Are you interested in seeing a detailed breakdown of the standard deviation calculation?
3. Do you need help with rounding intermediate steps to ensure accuracy?
4. Would you like a refresher on interpreting confidence intervals?
5. Are there any other confidence levels you’d like to explore for comparison?

**Tip:** When calculating a confidence interval, always make sure your sample size and degrees of freedom align with the appropriate critical value.

Two microprocessors are compared on a sample of 6 benchmark codes to determine if there is a difference in speed. Calculate a 98% confidence interval for the difference between the mean speeds of Processor A and Processor B.

Learn how to calculate a 98% confidence interval for the difference in mean speeds between two processors using benchmark code times. This tutorial covers statistical concepts such as mean differences, standard deviation, and critical t-values for accurate interval estimation.

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