What’s Wrong? Education Week magazine published a list consist...

Question

What’s Wrong? Education Week magazine published a list consisting of the mean teacher salary in each of the 50 states for a recent year. If we add the 50 means and then divide by 50, we get $56,479. Is the value of $56,479 the mean teacher salary for the population of all teachers in the 50 United States? Why or why not?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. An education survey published a list of the average student to teacher ratio in each of the 50 cities located in a country. If we add the 50 city averages and then divide by 50, we obtain a value of 16.2 students per teacher. Is this value of 16.2 considered the true average for the student to teacher ratio for all the classrooms located in the country? Why or why not? Awesome. So it appears for this particular problem, we're asked to solve for two separate things or two separate answers. Firstly, we're trying to figure out if this value of 16.2 is considered the true average for the student to teacher ratio for all the classrooms located in this particular country, and we're asked to say, why or why not, this is considered the true average. Awesome. So, like two separate things are technically one of the same. So now that we know what we're ultimately trying to solve, or we're trying to figure out this value of 16.2 is considered the true average, let's read off our multiple choice answers to see what our final answer might be. A is the average should be calculated by taking the median instead of the mean. B is student to teacher ratios cannot be average because they vary too much. C is the national average must be calculated as a weighted mean, not a simple average. And finally, D is the data includes only 50 cities and excludes smaller towns. Awesome. So, first off, we need to note that the key issue for this particular prong is whether the mean of the city level averages accurately represents will accurately represent the overall population mean. So considering that thought, our first step is we need to understand the calculation. So we need to interpret all the information that's provided to us and see how we're going to manipulate it to solve for a final answer. So we need to note and recall that the 16.2 students per teacher was obtained by averaging the report of city means. However, as we should note, this does not necessarily equal the true mean for all the classrooms nationwide. So, with that in mind, our second step now is we need to figure out why is this calculation incorrect. So why is it incorrect? So we need to consider the unequal city populations. We need to note that some cities may have many more classrooms than others. And a simple average treats all the cities equally, regardless of how many teachers or students they may have. We also need to consider a proper calculation method that could be implemented. So, as we should recall, in order to find the true national mean, we should use a weighted average where each city's contribution. Is based on the actual number of classrooms. So that would mean that it would ensure that larger cities will contribute proportionally more to the overall mean. So with that said, our third step now is we need to put together our final conclusions. So we need to recall a note that the value of 16.2 students per teacher is not the true average, so it's not the true average, so not true average. Awesome, so we need to make a note that, so not true average. So that would mean because this value of 16.2 students per teacher is not the true average for all the classrooms in this particular country, because, because this is very important, it does not account for the different city sizes because each city could be a different size or population. So, with that in mind, that would mean, without a doubt, that our final answer has to be a weighted mean. Considering the number of classrooms per city would give a more accurate result, so that would mean, without a doubt, the final answer has to be multiple choice answer C, which once again is the national average must be calculated as a weighted mean, not a simple average. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Mean vs. Weighted Mean

Population vs. Sample

Variability and Distribution

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