Burning Matches

73 Ohio Blue Tip Matches were randomly selected. For each match...

• it was placed in a vice in the upright position,
• a lighter was applied to the tip,
• at ignition (marked by a distinctive sound) a stopwatch was started,
• at death (marked by a wisp of smoke) the stopwatch was stopped, and
• the burn time was measured and recorded.

Our objective: obtain a 90% confidence interval ffor the population mean burn time.

Here's the data

47.26   60.17   69.75   58.13   63.67   62.84   63.80   62.16   43.63
61.56   70.60   25.29   76.33   57.64   68.32   87.87   30.21   62.14
61.07   58.10   59.97   27.84   59.84   67.82   63.12   64.29   70.15
56.74   61.80   63.24   35.87   35.58   42.02   74.13   63.83   72.41
32.56   22.39   40.25   50.25   65.52   41.34   70.07   67.23   71.91
76.56   68.71   30.15   52.30   64.97   59.03   67.54   63.29   49.92
63.52   71.27   48.32   57.35   63.96   74.69   66.54   53.96   60.84
74.99   80.37   68.12   62.55   62.69   40.34   72.13   19.85   59.16
66.20

Here's a histogram of the data along with some descriptive statistics.

The population mean burn time is denoted m. m is unknown because we don't have the time to burn every match.

To form a confidence interval use the expression

X-bar +/- z_a/2 [s / sqrt(n)]

• +/- stands for "plus or minus"
• sqrt stands for "square root of"
• X-bar is the sample mean.
• For large samples (say, n > 30 in most cases), the population standard deviation s may be replaced by the sample standard deviation S.

X-bar is the estimate, z_a/2 [s / sqrt(n)] is the margin of error.

1. Find the tabled value. The desired confidence level is 90%, the error rate is 0.10 or 10%. Split the error rate in two for errors in either direction, that's 0.05. We need z_0.05 = 1.645 (use the tables or the interactive tables on this site).
2. Compute the "standard error of the mean": S / sqrt(n) = 14.52/ sqrt(73) = 1.699 or 1.70. Notice that this number is provided in the table above. Many software packages compute it by default (because it's important).
3. Multiply the standard error by the tabled value to obtain the margin of error: 1.645 * 1.70 = 2.80. This number should be at the same decimal accuracy as is the estimate.
4. The interval is then represented in any of the following ways
   • 58.49 +/- 2.80.
   • Within 2.80 of 58.49.
   • Between 55.69 and 61.29.
   • (55.69, 61.29)

This is a 90% confidence interval for the population mean burn time m of all Ohio Blue Tip Kitchen Matches.

Statistical software will also produce this interval; use the "t-interval" commands. In Minitab there's also an option to this command that allows you to see the interval plotted along with the histogram. Minitab output and the histogram are shown below.

T Confidence Intervals

Variable	N	Mean	StDev	SE Mean	90.0% CI
Time	73	58.49	14.52	1.70	(55.66, 61.33)

The final result is slightly different than ours above. Technically this result is more accurate (it uses the "T" method rather than the "Z" method; however, the two methods given very similar results for large sample sizes).

Notice that it IS NOT THE CASE that 90% of the observations are in this interval. In fact, go back up to the data and figure out how many are between 55.69 and 61.33 -- it's only 12 of 73 which is about 15.4%. Note that the quartiles -- reported above -- are 51.28 and 67.97. About 50% of the data are between these values. Be careful: A confidence interval IS NOT a prediction interval. A prediction interval (PI) predicts a future observation; a confidence interval (CI) is an estimate of the population mean made on the basis of the sample mean and it's sampling variability.

We are 90% confident that the population mean burn time of all matches is between 55.69 and 61.33. What does this mean?

The procedure used to compute the interval includes the population mean in 90% of all samples.

We have no way of knowing whether or not the population mean is between 55.69 and 61.33. All we know is that the procedural reliability of this method is 90%. In 90% of all 90% confidence intervals the population mean lies within the bounds of the interval.

Exercises? There's a worksheet.