A random sample of size 81 is taken from a large population, measuring the time it takes to complete a driver’s license examination. The sample mean was found to be 43 minutes, and the sample standard deviation 5.71 minutes. Construct a 95% confidence interval around the sample mean.
95% = men ± 1.96 SEm
SEm = SD/√n
I'll let you do the calculations.
To construct a confidence interval around the sample mean, we can use the formula:
Confidence interval = sample mean ± (critical value) * (standard deviation / √sample size)
Here, we know that the sample mean is 43 minutes, the sample standard deviation is 5.71 minutes, and the sample size is 81.
To calculate the critical value, we need to determine the confidence level. In this case, the confidence level is 95%, which means we need to find the critical value corresponding to an alpha level of 0.05 (since the remaining 5% is divided equally between both tails of the distribution).
To find the critical value, we can look it up in the standard normal distribution table or use a statistical software. For a 95% confidence level, the critical value is approximately 1.96.
Now we can substitute the values into the formula:
Confidence interval = 43 minutes ± (1.96) * (5.71 minutes / √81)
Calculating the square root of the sample size (√81 = 9) and simplifying the calculation:
Confidence interval = 43 minutes ± (1.96) * (5.71 minutes / 9)
Confidence interval = 43 minutes ± (1.96) * (0.63 minutes)
Lastly, we can calculate the upper and lower bounds of the confidence interval:
Upper bound = 43 minutes + (1.96) * (0.63 minutes)
= 43 + 1.23
= 44.23 minutes
Lower bound = 43 minutes - (1.96) * (0.63 minutes)
= 43 - 1.23
= 41.77 minutes
Therefore, the 95% confidence interval around the sample mean is approximately (41.77 minutes, 44.23 minutes).