Confidence Intervals for Population Means Assuming Known Variance

Today we transitioned from looking at the sampling model of sample means to working with this model to construct confidence intervals. I first went through looking at subjective confidence intervals. This allows students to see that as the gain confidence their interval widens to capture more values. It also allows me to communicate how confident we are that the true mean lies within the interval created.

From here I moved to asking students to consider how we could make use of the central limit theorem and the sampling model. I used the class's sibling data and assumed that the class standard deviation was in fact equal to the population standard deviation. We created the sampling model and I drew a graph of this model on the board with the mean and standard deviations labeled above and below the mean. I then asked students what would be the 68.26% confidence interval? Once students grasped that it was just the interval from one standard deviation below to one standard deviation above the mean, they were much quicker about determining the 95.44% confidence interval. I hadn't labeled the graph beyond two standard deviations, so I asked them what the 99.74% confidence interval would be? Most students found the new end points although a few still had questions.

I pointed out that what we had done to construct the confidence interval was to add or subtract a integral multiple of the sampling model standard deviation away from the mean, i.e. μ ± Nσ where N = 1, 2, 3. This is all well and good but saying we are 95.44% confident seems a bit much; it would be nicer to have our confidence intervals at integral values rather than the multiples of the standard deviation.

I asked students what z-scores would result in a 95% confidence interval rather than a 95.44% confidence interval. This threw many of them for a loop. It was obvious that they still were not comfortable working  with normal models. After a bit more guidance students determined that z-scores of -1.96 and 1.96 are what were needed. I told the class the value of 1.96 was called a critical value as it was the z-score value that was needed to construct a 95% confidence interval.

For homework, I asked students to determine the critical values for a 90% and 99% confidence interval.

Below is the outline of today's lesson with italicized comments enclosed in square brackets [like this].

—Confidence Interval for One Population Mean

   ·         Confidence questions

o   Want students to realize as they gain confidence the spread of values increases

   ·         Take a sample, what is estimate of mean?

o   Use CLT to say best estimate is mean of sample

   ·         How can you account for sample variation?

o   Can the normal model help?

   ·         Every sample creates a different confidence interval [showed graphs of 20 confidence intervals developed from 20 samples, able to indicate that the interval may not contain the true population mean]

·         Calculating confidence intervals

o   Find z-scores, multiply by standard deviation, add and subtract from sample mean [focused on just adding and subtracting integral values and had students use practice problem below to first construct a 95.44% confidence interval]

o   Use calculator [decided to hold off on this until next class]

   ·         Practice – age of civilian workforce

o   Calculate 90% and 95% confidence intervals [used 95.44% confidence interval and then asked students to find 95% critical value, will continue this next class]