What is the appropriate formula to estimate the mean improvement in running time over two months using a 95% confidence interval?
To estimate the mean improvement in running time over two months using a 95% confidence interval, the appropriate formula to use is the t-confidence interval. In this case, the data consists of paired observations of initial and post times from the same individuals before and after participating in the new-runner program. The paired design of the data implies a dependency between the observations since each person's initial time is related to their post time.
The t-confidence interval formula takes into account this dependency by calculating the mean difference in running times, the standard error of the differences, and the t-critical value based on the degrees of freedom. The t-critical value is obtained from the t-distribution table, considering the desired confidence level (95%) and degrees of freedom (24 - 1 = 23). The t-critical value for a 95% confidence interval is approximately 2.068 (rounded to 3 decimal places).
What is the appropriate value for the t-critical point for this interval?
The appropriate value for the t-critical point for this interval is approximately 2.068 (rounded to 3 decimal places). This value is derived from the t-distribution table by considering the desired confidence level (95%) and degrees of freedom, which in this case is 23. The t-critical value plays a crucial role in calculating the confidence interval to estimate the mean improvement in running time for middle-aged adults who use the program for two months.
Calculate the confidence interval for the mean improvement in running time over two months using a 95% confidence level.
Calculating Confidence Interval:
Lower limit = 4.24 - (2.068 * 0.06) ≈ 3.98
Upper limit = 4.24 + (2.068 * 0.06) ≈ 4.50
The confidence interval for the mean improvement in running time over two months with a 95% confidence level is estimated to be between 3.98 and 4.50 minutes. This interval provides insight into the potential range of benefits offered by the new-runner program in terms of reducing running times for middle-aged adults.