To construct a 90% confidence interval estimate for the salaries of data scientists, we follow these steps:
1. Identify the given data:
- Sample size ([tex]\( n \)[/tex]): 20
- Sample mean ([tex]\( \bar{x} \)[/tex]): 800
- Sample standard deviation ([tex]\( s \)[/tex]): 238
- Confidence level: 90%
2. Determine the critical value:
- For a 90% confidence interval, the significance level ([tex]\( \alpha \)[/tex]) is: [tex]\( \alpha = 1 - 0.90 = 0.10 \)[/tex].
- Since we are working with a t-distribution (because the sample size is small, [tex]\( n < 30 \)[/tex]), we need to find the t-critical value for [tex]\( \alpha/2 = 0.05 \)[/tex] with [tex]\( df = n - 1 = 19 \)[/tex].
3. Calculate the t-critical value:
- The t-critical value for 19 degrees of freedom at [tex]\( \alpha/2 = 0.05 \)[/tex] is approximately 1.729.
4. Compute the margin of error (MOE):
- The margin of error is given by:
[tex]\[
\text{MOE} = t_{\text{critical}} \times \frac{s}{\sqrt{n}}
\][/tex]
- Substituting the values:
[tex]\[
\text{MOE} = 1.729 \times \frac{238}{\sqrt{20}} \approx 92.02
\][/tex]
5. Determine the confidence interval:
- The confidence interval is calculated as:
[tex]\[
\text{CI} = \left( \bar{x} - \text{MOE}, \bar{x} + \text{MOE} \right)
\][/tex]
- Substituting the values:
[tex]\[
\text{CI} = \left( 800 - 92.02, 800 + 92.02 \right) = \left( 707.98, 892.02 \right)
\][/tex]
6. Compare the calculated confidence interval with the given options:
- The calculated confidence interval [tex]\( (707.98, 892.02) \)[/tex] is very close to option (d), [tex]\( (707.99, 892.01) \)[/tex].
Therefore, the correct answer is:
d. [tex]\( (707.99 ; 892.01) \)[/tex]
