Confidence Intervals for Means

Another way to approach the CD running time data:

would be to estimate the actual mean running time of the 700+ CDs in the population. We could do this with a confidence interval, just as we used a one-proportion confidence interval to estimate the value of a population proportion.

As with the related hypothesis test we conducted in the previous section, we will need to use a t-distribution in place of a Normal (z) distribution, so we will call this confidence interval a t-interval. Let's construct a 95% confidence interval for the true mean running time of all 700+ CDs.

The conditions are the same as for the t-test: The CDs were randomly selected and only represent about 2% of the population, so it's reasonable to assume that the trials are independent, but the histogram, boxplot and Normal probability plot of the running times:

indicate that the data may not come from a population that follows a Normal distribution, which is a problem because our sample size is relatively small (n = 14). We will proceed with constructing the confidence interval to provide an example, but should be cautious about how we use the interval, since the conditions may not be satisfied.

As before, we can compute the mean (`bar y_1` = 63.6 minutes) and standard deviation (`s_1` = 10.0 minutes) of the 14 CD times in our one sample using Data Desk or 1-VarStats on the TI-84. We would like to compute the standard deviation of all possible sample means:

`SD(bar y) = (sigma)/(sqrt(n))`

but we don't know `sigma` (the standard deviation of the running times for all 700+ CDs) so instead we use the estimate:

`SE(bar y) = (s)/(sqrt(n)) = (10.0)/(sqrt(14)) approx 2.67`

In our one-proportion intervals, the margin of error for our 95% confidence interval was given by:

`ME = z text(*) times SE(hat p)`

Here, we might expect that the margin of error would be given by:

`ME = z text(*) times SE(hat y)`

except, as with the hypothesis test, we must use the t-distribution, so we need to use this formula

`ME = t text(*) times SE(bar y)`

instead, where `t text(*)`  is the critical value that separates the middle 95% of the t-distribution (with df = 13) from the bottom 2.5% and the top 2.5%.

We can compute this on the TI-84 using invT(0.975,13) = 2.160. This works just like invNorm did to compute z*, except in this case we also need to specify the number of degrees of freedom. The only problem is that there is no invT feature on the TI-83 (nor on the TI-84, unless you have the more recent operating system installed). If we can compute t*, then we can compute the margin of error:

`ME = t text(*) times SE(bar y) = 2.160 times 2.67 approx 5.8` minutes

We add and subtract this margin of error from the observed sample mean (63.6 minutes) to get a confidence interval of (57.8, 69.4), or:

`57.8 < mu < 69.4`

We are 95% confident that the true mean duration of these CDs is between 57.8 minutes and 69.4 minutes. However, we should be cautious about using these results, as the normality condition may have been violated. In this case it won't be a matter of life and death if my iPod doesn't hold quite as many CDs as I expect it to, but if we were estimating something for which there might be more dire consequences if we made a mistake, we may not want to report any results at all, or gather more data so that the normality assumption is not as vital (for sample sizes greater than 40 this condition isn't as crucial, but for sample sizes smaller than 15 it definitely is).

t-intervals on the TI-84
So what can we do if we don't have the invT function on the calculator? We can use the TInterval feature instead. If you entered the CD data into L1 to compute the mean and standard deviation in our previous computations, select the Data option, specify L1 for List, leave Freq set to 1, specify 0.95 for C-Level and then move the cursor to Calculate and press ENTER. If you did not enter the data into the calculator already, but you do know the summary statistics, select the Stats option, specify 63.6 for x, 10.0 for Sx, 14 for n and 0.95 for C-Level, then move the cursor to Calculate and press ENTER. Either way you should see the confidence interval like (57.826,69.374) displayed, which agrees with out previous computation.

While the TI-84 is useful for computing the confidence interval limits, it doesn't check assumptions and conditions, nor does it properly interpret the confidence interval. On an exam, you can use TInterval to compute the confidence interval limits, but be sure to remember the conditions and interpretation as well.

Exercises

1. Guessing ages A statistics student from Vietnam remembered a game show from that country called Guessing My Age. She decided to try something similar, asking an 18-year-old friend who still lives in Vietnam to send her a picture and then asking 50 Edmonds Community College students to guess the age of the person in the picture. The results are shown below:

2|6
2|44445
2|22233333
2 000011111111
1|888888899999
1|666777777
1|555               Key: 2|6 means 26 years old

Construct a 90% confidence interval for the mean guessed age of the friend by all EdCC students.

2. Handwashing A 2009 WHO report noted that among studies related to health care workers washing their hands to prevent infection, one problem is "the duration of  hand treatments[, which] require subjects to treat their hands with the hand hygiene product or a positive control for 30 seconds or 1 minute, despite the fact that the average duration of  hand cleansing by HCWs has been observed to be less than 15 seconds in most studies." A registered nurse at a Seattle-area hospital, observed 21 health care workers in the Special Care Unit (SCU) and recorded the number of seconds that each employee washed his or her hands before entering a treatment room. The times appear in the display below:

1|67
1|44
1|222233
1|00001
0|88889
0|6        Key: 1|7 = 17 seconds

Construct a 95% confidence interval for the mean amount of time spent washing hands by health care workers in the SCU at this hospital.