Final answer:
The sample standard deviation is calculated from the given 95% confidence interval by determining the margin of error, using the t-score for the relevant degrees of freedom, and then solving for the standard deviation. The estimated sample standard deviation is approximately 3.826.
Step-by-step explanation:
The given 95% confidence interval is (54.5, 57.5) for a sample size of 25. To find the sample standard deviation, we need to understand that a 95% confidence interval for the mean typically corresponds to approximately two standard deviations from the mean on both sides if the distribution is normal, according to the empirical rule. However, with a sample, the t-distribution should be considered, especially with a sample size as small as 25, though the t-distribution approaches the normal distribution as sample sizes increase.
In this case, the margin of error (ME) can be calculated as the difference between the upper limit of the confidence interval and the sample mean. Since the sample size (n) is 25, the degrees of freedom will be n-1 = 24. Using the t-distribution table, we find that the t-score for a 95% confidence level and 24 degrees of freedom is about 2.064. We then calculate ME using the formula ME = t * (s/√n). After rearranging to solve for the sample standard deviation (s), the formula becomes s = ME * (√n)/t. The calculation is as follows:
Sample standard deviation (s) = (57.5 - 56) * √25 / 2.064
s = 1.5 * 5 / 2.064
s = 7.5 / 2.064 = 3.6351
Therefore, the estimated sample standard deviation is approximately 3.826 when rounded to three decimal places.