Answer :
To calculate the 95% confidence interval for a sample mean, given a sample mean of 20, a sample standard deviation of 10, and a sample size of 9, follow these steps:
1. Determine the sample mean and the sample size:
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 20
- Sample size ([tex]\( n \)[/tex]) = 9
2. Determine the sample standard deviation:
- Sample standard deviation ([tex]\( s \)[/tex]) = 10
3. Calculate the standard error of the sample mean:
[tex]\[ \text{Standard Error} = \frac{s}{\sqrt{n}} = \frac{10}{\sqrt{9}} = \frac{10}{3} \approx 3.33 \][/tex]
4. Determine the z-score for a 95% confidence level:
- The z-score corresponding to a 95% confidence level is approximately 1.96. This value comes from standard normal distribution tables.
5. Calculate the margin of error:
[tex]\[ \text{Margin of Error} = z \times \text{Standard Error} = 1.96 \times 3.33 \approx 6.53 \][/tex]
6. Determine the lower and upper bounds of the confidence interval:
- Lower bound: [tex]\( \bar{x} - \text{Margin of Error} = 20 - 6.53 = 13.47 \)[/tex]
- Upper bound: [tex]\( \bar{x} + \text{Margin of Error} = 20 + 6.53 = 26.53 \)[/tex]
Thus, the 95% confidence interval for the sample mean is [tex]\( (13.47, 26.53) \)[/tex].
Therefore, the correct answer is:
- 13.47 to 26.53
1. Determine the sample mean and the sample size:
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 20
- Sample size ([tex]\( n \)[/tex]) = 9
2. Determine the sample standard deviation:
- Sample standard deviation ([tex]\( s \)[/tex]) = 10
3. Calculate the standard error of the sample mean:
[tex]\[ \text{Standard Error} = \frac{s}{\sqrt{n}} = \frac{10}{\sqrt{9}} = \frac{10}{3} \approx 3.33 \][/tex]
4. Determine the z-score for a 95% confidence level:
- The z-score corresponding to a 95% confidence level is approximately 1.96. This value comes from standard normal distribution tables.
5. Calculate the margin of error:
[tex]\[ \text{Margin of Error} = z \times \text{Standard Error} = 1.96 \times 3.33 \approx 6.53 \][/tex]
6. Determine the lower and upper bounds of the confidence interval:
- Lower bound: [tex]\( \bar{x} - \text{Margin of Error} = 20 - 6.53 = 13.47 \)[/tex]
- Upper bound: [tex]\( \bar{x} + \text{Margin of Error} = 20 + 6.53 = 26.53 \)[/tex]
Thus, the 95% confidence interval for the sample mean is [tex]\( (13.47, 26.53) \)[/tex].
Therefore, the correct answer is:
- 13.47 to 26.53
