Science:Math Exam Resources/Courses/MATH102/December 2012/Question B 03

MATH102 December 2012

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Question B 03

A researcher measures the length of 16 adult wombats and finds a sample mean of 57 cm and a sample standard deviation of 12 cm. The standard error of the mean is estimated to be 3 cm. If she plans to carry out measurements on 100 more wombats, the following can be expected (place an X in exactly one column for each row): decrease stay roughly the same increase The sample mean will The sample maximum will The standard error of the mean will

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

Each row of the table can be stated as its own question, most of which can be determined through common sense: If you randomly sample a population, should its mean change depending on the size of the sample? If you were sampling a group of people, which would have the tallest person: a large group or small group? The standard error of the mean reflects the difference between the sample mean and the "true" population mean. As you increase the sample size, will this bring the sample mean closer to the population mean, farther away, or the same distance?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let us consider each row of the table: Sample Mean The sample mean should stay roughly the same. This is because the mean length of adult wombats is a well-defined quantity, and we have no information to suggest that we have been sampling adult wombat length in a biased manner. Sample Maximum The sample maximum is likely to increase. This is simply because the larger the set sampled, the more likely you are to include extreme outliers, which will increase the maximum and decrease the minimum of your set. In more technical notation, this is the same as saying $\max _{A}x\geq \max _{B}x$ when B is a subset of A. Standard Error of the Mean The standard error of the mean will decrease. Recall that the standard error of the mean estimates how far away your sample mean is from the "true" population mean. More and more samples will improve the accuracy of your sample mean, decreasing the error. This can also be seen in the formula for standard error of the mean: ${\text{SE}}\ ={\frac {s}{\sqrt {n}}}$ where s is the standard deviation of the sample and n is the number of objects sampled. As n increases, the overall fraction will decrease. So the table should be filled in as follows: decrease stay roughly the same increase The sample mean will X The sample maximum will X The standard error of the mean will X

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