To construct a 95% confidence interval for the sample means provided, we need to follow several steps. Here is the detailed solution:
1. Calculate the Sample Mean:
The sample mean ([tex]\(\bar{x}\)[/tex]) is calculated by summing all the sample means and dividing by the number of samples.
Given the data: [tex]\(\{4, 4, 4, 4.2, 4.2, 4.3, 4.3, 4.3, 4.4, 4.4, 4.4, 4.4, 4.5, 4.5, 4.6, 4.7, 4.7, 4.7, 4.8, 4.8, 4.8, 4.9, 4.9, 4.9, 4.9\}\)[/tex]
The sample mean [tex]\(\bar{x}\)[/tex] is approximately [tex]\(4.504\)[/tex].
2. Calculate the Sample Standard Deviation:
The sample standard deviation (s) measures the spread of the sample means from the sample mean.
Given that the sample standard deviation is approximately [tex]\(0.29788\)[/tex].
3. Calculate the Standard Error:
The standard error (SE) is the sample standard deviation divided by the square root of the sample size (n).
Given that the standard error is approximately [tex]\(0.05958\)[/tex].
4. Determine the Critical Value:
For a 95% confidence interval, the critical value can be found using the t-distribution because the sample size is relatively small. The degrees of freedom (df) equals [tex]\(n-1\)[/tex].
Given that the critical value (for a 95% confidence level) is approximately [tex]\(2.0639\)[/tex].
5. Calculate the Margin of Error:
The margin of error (MOE) is the critical value multiplied by the standard error.
Given that the margin of error is approximately [tex]\(0.12296\)[/tex].
6. Construct the Confidence Interval:
The confidence interval is calculated by subtracting and adding the margin of error from the sample mean.
- The lower bound of the confidence interval is [tex]\(\bar{x} - \text{MOE}\)[/tex].
- The upper bound of the confidence interval is [tex]\(\bar{x} + \text{MOE}\)[/tex].
Given that the lower bound is approximately [tex]\(4.3810\)[/tex] and the upper bound is approximately [tex]\(4.6270\)[/tex].
Therefore, the 95% confidence interval for the sampling distribution's mean is approximately [tex]\((4.3810, 4.6270)\)[/tex].
By following the steps above, we have successfully constructed a 95% confidence interval for the given sample means.
