## Question 698:

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:

Part a.

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. Part b. To have a margin of error of +/- .03 grams, assuming your standard deviation stays the same and only the sample size increases will affect the standard error of the mean, you just need to setup an equation to solve for the unknown sample size.
1. We know the margin of error is the standard error of the mean SEM * a critical t or z value. Because we don't know the sample size, which we need to use the t-statistic, we'll need to use the z statistic here or iterate through some t-values until we converge on the correct one. For a 90% level of confidence, the 2-sided critical z-value is 1.644 and I'll walk through that first (as it will also serve as our initial best guess for the second approach).
2. Working backwards from a confidence interval we first divide the desired margin by the critical value = .03/1.644= .018239 (SEM)
3. Next we divide the standard deviation by the result from the previous step = .1319/.018239 =7.231873.
4. Finally, we square the result from the last step = 52.2999
5. Rounding up gets us a sample size of 53 which will generate a margin of error of  +/- .03 grams with 90% confidence.
6. Optional: Now that we have a sample size to work with we can use this for the iterative t-statistic approach. The t-statistic on 52 degrees of freedom for a 90% level of confidence is 1.674689 (which can be found using the  Inverse t-Distribution Calculator. Compare this with the z of 1.644 and we see it's a bit larger. This gives us a margin of error of .018239*1.674689 = .30342, which is a bit higher than .03. If we increase the sample size to 55 we get a t-statistic of 1.673565 and the margin is .029765. So the sample size using this method is 55 (2 higher than the z method). Since the sample size was above 30 in this example, there won't be much difference in methods. When the sample is smaller, there will be a bigger difference and the t-iteration method is recommended. The one you pick to use for this example depends on what you've learned so far and your desired level of precision.

Part c
Variation on the weight of the tootsie rolls could be cause by slight defects in the molds or aberrations in the extruder which pushes the mix into the molds. The cutting and cooling process can also lead to irregularities as well as temperature, humidity and foreign matter in the mix. The calibration of the weighing machine also might not be as sensitive as the scale used in this example, resulting in less precise weights.