To find a 90% confidence interval for the mean reaction times based on the sample, we'll follow these steps:
- Calculate the sample mean (\(\bar{x}\)).
- Calculate the sample standard deviation (\(s\)).
- Determine the critical t-value for a 90% confidence interval.
- Calculate the margin of error.
- Construct the confidence interval.
Step 1: Calculate the Sample Mean (\(\bar{x}\))
The reaction times are:
- 3.0, 4.0, 3.1, 5.0, 2.9, 2.3, 3.0, 4.1
\[ \bar{x} = \frac{\sum x}{n} = \frac{3.0 + 4.0 + 3.1 + 5.0 + 2.9 + 2.3 + 3.0 + 4.1}{8} \]
\[ \bar{x} = \frac{27.4}{8} = 3.425 \]
Step 2: Calculate the Sample Standard Deviation (\(s\))
Calculate the deviations from the mean, square them, sum them, and then compute \(s\):
\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \]
Calculating each deviation:
- \( (3.0 - 3.425)^2 = 0.180625 \)
- \( (4.0 - 3.425)^2 = 0.330625 \)
- \( (3.1 - 3.425)^2 = 0.105625 \)
- \( (5.0 - 3.425)^2 = 2.505625 \)
- \( (2.9 - 3.425)^2 = 0.275625 \)
- \( (2.3 - 3.425)^2 = 1.265625 \)
- \( (3.0 - 3.425)^2 = 0.180625 \)
- \( (4.1 - 3.425)^2 = 0.455625 \)
Summing them up:
\[ \sum (x_i - \bar{x})^2 = 0.180625 + 0.330625 + 0.105625 + 2.505625 + 0.275625 + 1.265625 + 0.180625 + 0.455625 = 5.025 \]
Now calculate \(s\):
\[ s = \sqrt{\frac{5.025}{8 - 1}} = \sqrt{\frac{5.025}{7}} \approx \sqrt{0.717857} \approx 0.847 \]
Step 3: Determine the Critical t-value
For a 90% confidence interval with \(n - 1 = 8 - 1 = 7\) degrees of freedom, we can use the t-distribution table or calculator to find the critical t-value (\(t_{\alpha/2}\)).
Using a t-table for 7 degrees of freedom at the 0.05 significance level:
\[ t_{0.05, 7} \approx 1.895 \]
Step 4: Calculate the Margin of Error
The margin of error (ME) is calculated as follows:
\[ \text{ME} = t_{\alpha/2} \times \frac{s}{\sqrt{n}} = 1.895 \times \frac{0.847}{\sqrt{8}} \approx 1.895 \times 0.299 = 0.567 \]
Step 5: Construct the Confidence Interval
Finally, we can construct the confidence interval as follows:
\[ \text{Confidence Interval} = \left( \bar{x} - \text{ME}, \bar{x} + \text{ME} \right) = \left( 3.425 - 0.567, 3.425 + 0.567 \right) \]
\[ \text{Confidence Interval} = \left( 2.858, 3.992 \right) \]
Conclusion
The 90% confidence interval for the mean of all reaction times is approximately \((2.858, 3.992)\).