An article investigated the consumption of caffeine among women. A sample of 53 women were asked to monitor their caffeine intake over the course of one day.

Question

The mean amount of caffeine consumed in the sample of women was 232.082 mg with a standard deviation of 217.129 mg. In the article, researchers would like to include a 99% confidence interval. With 99% confidence, we estimate the mean amount of caffeine consumed by women is between ? mg and ? mg.

Answer 1

(A quick note before I start here: I almost submitted this response with a z-value for the critical value! Before I could submit it, I changed it to a t-value. Remember to take a second to make the correct decision about using the z- or t-distribution for your critical value before doing problems like this.)

Let sample size n be 53, let sample mean x-bar be 232.082, and let sample standard deviation s be 217.129.

Since the population mean and population standard deviation are unknown, we get our critical value from the t-distribution, and it will be with n - 1 = 53 - 1 = 52 degrees of freedom. Let your critical value be t_α/2 = t_0.01/2 = t_0.005 = 2.674.

(This comes from the fact that we're using 99% confidence, so 1% of area is left under the curve. This is turned into a decimal of 0.01 and cut in half. Then you find what t-value with 53 df has 0.005 area to its right. Also, if you're limited to a printed t-table that gives you rows for 50 and 60 df, defer to the row with 50 df. In which case, use 2.678. I use the more precise 2.674 here.)

Then your confidence interval will be set up this way:

x-bar ± t_α/2(s/√n)

232.082 ± (2.674)(217.129/√53)

232.082 ± (2.674)(29.825)

232.082 ± 79.752

Find the lower bound 232.082 - 79.752.

Find the upper bound 232.082 + 79.752.

We get an interval of (152.33, 311.83).

(Running this TInterval procedure in a TI-83 or TI-84 gets you an interval of (152.34, 311.83). This is merely a matter of accuracy and rounding.)

With 99% confidence, we estimate the mean amount of caffeine consumed by women is between 152.33 mg and 311.83 mg.

Another way to interpret this interval is to say that we are 99% confident that the mean amount of caffeine consumed by women is between 152.33 mg and 311.83 mg.