## Question 354:

1

Since this sample size is relatively small (10) I recommend building your confidence interval using the t-statistic. Assuming the data is roughly normally distributed:

1. The mean and standard deviation are 3.305 and .1319 respectively for the tootsie-rolls.
2. Calculate the standard error of the mean: standard deviation / square root of the sample size = .1319/SQRT(10) = .0417
3. Find the critical value from the t-distribution for 9 degrees of freedom and a probability of .10 (since this is for a 90% confidence interval). You can use MS Excel and type =TINV(.10,9) and should get 1.833.
4. Calculate the margin of error: which is the standard error of the mean times the t-value = .0417*1.833 = .0764
5. Generate the Lower and Upper bounds of the confidence interval by adding and subtracting the margin of error to the mean = 3.305 +.0764  = 3.3814 and = 3.305 -.0764  =3.2286.
6. So your 90% confidence interval around the mean is (3.228, 3.381).
7. To have a margin of error of ± 3 assuming your standard deviation stays the same and only the sample size increases will affect the standard error of the mean. While there are some complicated ways to calculate sample size, an easy approach is trial and error. Change the sample size in the equations above until the margin is around .03. I found a sample of around 54 to provide a margin of .030069.