73 Ohio Blue Tip Matches were randomly selected. For each match...
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.
Descriptive Statistics | |
Number of observations | 73 |
Mean | 58.49 |
Median | 62.55 |
Trimmed mean | 59.29 |
Standard deviation | 14.52 |
Standard error of the mean | 1.70 |
Minimum | 19.85 |
Maximum | 87.87 |
First quartile | 51.28 |
Third quartile | 67.97 |
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)]
X-bar is the estimate, z_{a}_{/2} [s / sqrt(n)] is the margin of error.
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.