To find the 99% confidence interval for the mean completion time, let's go through the steps required, as the sample data is already provided. Here’s the approach we’ll use:

1. Calculate the Sample Mean ($\bar{x}$): Add up all the completion times and divide by the number of subjects (14).

2. Calculate the Sample Standard Deviation (s):

   • Find each score's deviation from the mean.
   • Square each deviation.
   • Sum these squared deviations, divide by $n - 1$ (where $n = 14$), and then take the square root.

3. Determine the T-Score for 99% Confidence Level:

   • Since $n = 14$, we have 13 degrees of freedom ($n - 1$).
   • Using a t-table or calculator, find the t-score corresponding to 99% confidence with 13 degrees of freedom.

4. Calculate the Margin of Error (ME): $ME = t \times \frac{s}{\sqrt{n}}$

5. Construct the Confidence Interval:

   • Lower limit = $\bar{x} - ME$
   • Upper limit = $\bar{x} + ME$

Let's go through the calculations in detail:

1. Sample Mean ($\bar{x}$)

Completion times: $32, 38, 43, 49, 45, 27, 47, 33, 39, 43, 44, 32, 49, 26$

$\bar{x} = \frac{32 + 38 + 43 + 49 + 45 + 27 + 47 + 33 + 39 + 43 + 44 + 32 + 49 + 26}{14}$

2. Sample Standard Deviation (s)

Calculate the deviations from the mean, square each deviation, sum them, and divide by $n - 1$, then take the square root.

3. T-Score for 99% Confidence with 13 Degrees of Freedom

From the t-table, look up the t-value for 99% confidence and 13 degrees of freedom.

4. Margin of Error (ME)

$ME = t \times \frac{s}{\sqrt{14}}$

5. Confidence Interval

Once we have the margin of error, we can construct the confidence interval:

$\text{Lower limit} = \bar{x} - ME$ $\text{Upper limit} = \bar{x} + ME$

Let me compute these values for you.The calculations yield the following results for the 99% confidence interval:

• Sample Mean ($\bar{x}$): 39.1 seconds (rounded to one decimal place)
• Sample Standard Deviation (s): 7.9 seconds
• T-Score for 99% Confidence with 13 Degrees of Freedom: 3.012
• Margin of Error (ME): 6.3 seconds

Confidence Interval

• Lower limit: 32.7 seconds (rounded to one decimal place)
• Upper limit: 45.4 seconds (rounded to one decimal place)

Therefore, the 99% confidence interval for the true mean completion time is: $(32.7, 45.4)$

Would you like further details on these calculations or have any questions?

Here are five related questions to deepen your understanding:

1. What would happen to the confidence interval if the confidence level were changed to 95%?
2. How does increasing the sample size affect the width of the confidence interval?
3. What role does the t-score play in the calculation of the confidence interval?
4. How would the interval change if we used a larger sample standard deviation?
5. Why do we use the t-distribution instead of the normal distribution in this calculation?

Tip: Always check the degrees of freedom when using the t-distribution, as it directly affects the t-score value.