Estimating a Mean (Lesson 8.5)
Chapter 8 - Day 7
State and check the Random and Normal/Large Sample conditions for constructing a confidence interval for a population mean.
Determine critical values for calculating a C% confidence interval for a population mean.
Calculate a C% confidence interval for a population mean.
For this lesson and the next, we move away from the world of proportions into the world of means! Several things will change in this new world, including conditions for inference, formulas, and a new critical value (t*).
Students will use their prior knowledge from Chapter 7 to work through questions #1-5 in their groups. The teacher must then STOP and do a debrief of questions #1-5, then take the whole group through question #6. This is where the rationale for using the new t* critical value is explained (see below) and students are shown how to use Table B. Once equipped with t*, students can complete the rest of the activity in their groups.
In question #5 in the activity, students are asked to calculate the standard deviation of the sampling distribution of a sample mean. This is actually impossible because they would need to know the true standard deviation of the population of all students, which is likely unknown. But they do have a good guess - the sample standard deviation from their sample of 30 students. So now students are actually just estimating the standard deviation, which is why we now call this the standard error.
For question #6 in the activity, provide a short explanation as to why we use t* instead of z*. When trying to estimate the mean of a population, we also have to estimate the standard deviation of the population in order to understand the standardized distribution of sample means. Because we are now trying to estimate 2 parameters, we are introducing more variation into the standardized statistic. Thus the standardized distribution of sample means follows a t-distribution, which is wider and shorter than a normal distribution.
Then show students how to use Table B to find t* critical values. Remind them that if the df they are looking for in the table is not available, to use the next lowest df that is available. This is the conservative approach, as it will give them a slightly higher t* (and slightly wider confidence interval) than is actually needed. You might also point out that the z* critical values appear at the bottom of Table B (when df = infinity the t-distribution turns into a normal distribution).
Notice in the QuickNotes that we are giving 2 versions of the formula for a confidence interval for a mean. We want students to get in the habit of writing these two formulas before they plug numbers in to get an answer. We call them the "General formula" and the "Specific formula". The nice thing about the general formula is that is is the same for any confidence interval we will calculate in this course.