A random sample of 36 statistics students were asked to add a column of numbers. The amount of time required to compute the sum was recorded for each student. The sample mean time was 54 seconds with a sample standard deviation of 18 seconds.
1. Construct a 95% confidence interval for mu, the mean time for the population of statistics students, to solve those problems
95% = mean ± 1.96SEm
SEm = SD/√n
To construct a confidence interval for the mean time, we will use the formula:
Confidence Interval = Sample Mean ± (Critical Value) * (Standard Error)
Step 1: Find the critical value
The critical value depends on the desired confidence level and the sample size. In this case, we want a 95% confidence level. Since the sample size is 36, we need to find the critical value from the t-distribution table.
The degrees of freedom (df) for a t-distribution is equal to the sample size minus 1. In this case, df = 36 - 1 = 35.
Looking up the critical value for a 95% confidence level and 35 degrees of freedom, we find it to be approximately 2.028.
Step 2: Calculate the standard error
The standard error (SE) measures the variability of the sample mean. It is calculated using the formula:
Standard Error = Sample Standard Deviation / √(Sample Size)
In this case, the sample standard deviation is 18 seconds and the sample size is 36. Plugging in these values, we get:
Standard Error = 18 / √(36) = 18 / 6 = 3.
Step 3: Calculate the confidence interval
Now we can plug the values into the formula to calculate the confidence interval.
Confidence Interval = Sample Mean ± (Critical Value) * (Standard Error)
Confidence Interval = 54 ± (2.028) * (3)
Confidence Interval = 54 ± 6.084
Confidence Interval = (47.916, 60.084)
Therefore, the 95% confidence interval for the mean time for the population of statistics students to solve those problems is (47.916 seconds, 60.084 seconds).