To calculate the 90% confidence interval for a sample mean of 15 with a sample standard deviation of 5 and a sample size of 25, follow these steps:
1. Identify the sample mean ([tex]\(\bar{x}\)[/tex]):
- Given: [tex]\(\bar{x} = 15\)[/tex]
2. Identify the sample standard deviation (s):
- Given: [tex]\(s = 5\)[/tex]
3. Identify the sample size (n):
- Given: [tex]\(n = 25\)[/tex]
4. Determine the confidence level:
- Given: 90%
5. Find the critical z-value (z) for a 90% confidence level:
- The 90% confidence level corresponds to 0.90. For a two-tailed test, we split the 0.10 tail area evenly, giving us 0.05 in each tail. Using standard statistical tables or a calculator, the critical z-value for 90% confidence is approximately 1.645.
6. Calculate the standard error of the mean (SE):
- Formula: [tex]\(SE = \frac{s}{\sqrt{n}}\)[/tex]
- Calculation: [tex]\(SE = \frac{5}{\sqrt{25}} = \frac{5}{5} = 1\)[/tex]
7. Calculate the margin of error (ME):
- Formula: [tex]\(ME = z \times SE\)[/tex]
- Calculation: [tex]\(ME = 1.645 \times 1 = 1.645\)[/tex]
8. Calculate the lower and upper bounds of the confidence interval:
- Lower bound: [tex]\(\bar{x} - ME = 15 - 1.645 \approx 13.36\)[/tex]
- Upper bound: [tex]\(\bar{x} + ME = 15 + 1.645 \approx 16.64\)[/tex]
Therefore, the 90% confidence interval for the sample mean is approximately [tex]\(13.36\)[/tex] to [tex]\(16.64\)[/tex].
Comparing with the given options:
- 12.94 to 17.06
- 13.29 to 16.71
- 14.66 to 15.34
- 13.36 to 16.65
The correct interval, accurate to the nearest decimal, is [tex]\(13.36\)[/tex] to [tex]\(16.65\)[/tex].
