Final answer:
To identify the symmetrical interval that includes 95% of the sample means, calculate the standard error of the mean (SEM), find the Z-score corresponding to a 95% confidence level, and use the formula lower bound = mean - (SEM * Z-score) and upper bound = mean + (SEM * Z-score). For this population with a mean of 55 and standard deviation of 14, the interval is (51.08, 58.92).
Step-by-step explanation:
To identify the symmetrical interval that includes 95% of the sample means, we can use the Z-distribution table. Since the population mean is 55 and the standard deviation is 14, we need to calculate the standard error of the mean (SEM) first. The formula for SEM is SEM = standard deviation / sqrt(sample size). Therefore, SEM = 14 / sqrt(49) = 2.
Next, we look at the Z-distribution table and find the corresponding Z-score for a 95% confidence level. The Z-score for a 95% confidence level is approximately 1.96.
To calculate the lower bound of the interval, we subtract the product of SEM and Z-score from the population mean: lower bound = 55 - (2 * 1.96) = 55 - 3.92 = 51.08.
To calculate the upper bound of the interval, we add the product of SEM and Z-score to the population mean: upper bound = 55 + (2 * 1.96) = 55 + 3.92 = 58.92.
Therefore, the symmetrical interval that includes 95% of the sample means is (51.08, 58.92).