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Text Sections 9.4, 9. So far we have seen how to think rationally about conducting tests for the various aspects of a population that we care about. Since most people like to have a quick summary of a large data set (for example, what is the median selling price for a home in your neighborhood?) we like to be able to obtain useful information from our sample regarding population parameters such as the mean and the standard deviation.
At this point you should feel reasonably comfortable with our methodology for conducting a test (the 6 steps) and how to implement this methodology for several particular cases. In particular, you should know how to ๏ท Conduct a test about the population mean ๐ when you have a large sample for reasonably shaped populations and when you have a small sample in the special case of a normal population. Remember that our class does not discuss small samples from non-nomral populations. If your population has a good deal of skew you need to think carefully about how to proceed.
Large sample statistic when testing ๐ : ๐ง = ๐ฅ โ๐ ๐ ๐
Small sample statistic when testing ๐ : ๐ก = ๐ฅ โ๐ ๐ ๐
๏ท You know how to conduct a test about a population proportion.
Statistic when testing ๐ : ๐ง = ๐ โ^ ๐ ๐ 1 โ ๐ ๐
๏ท When you have a question about whether two populations have the same mean and you obtain independent samples you know how to use the statistic
Small sample statistic when testing ๐ 1 โ ๐ 2 : ๐ก = ๐ฅ^1 โ ๐ฅ^2 โ^ ๐^1 โ ๐^2 ๐ 12 ๐ 1 +^
๏ท If you have a question about whether two populations have the same mean and you obtain independent samples you can use the following statistic if your samples are both at least of size 30:
Large sample statistic when testing ๐ 1 โ ๐ 2 : ๐ง = ๐ฅ^1 โ ๐ฅ^2 โ^ ๐^1 โ ๐^2 ๐ 12 ๐ 1 +^
So, whatโs left to do? Your book tells you how to conduct tests concerning a populationโs standard deviation, and how to compare standard deviations from two populations. In this introductory class we arenโt going to worry about that. You may want to compare proportions from two populations. For example, you might want to test whether men and women have high blood pressure at the same rates. In this case you would want to test whether ๐๐๐๐ = ๐๐ค๐๐๐๐ for example. You also might want to test for a difference in population means when your data come from samples which are not independent (for instance, to see whether an SAT preparation course adds value you could round up 40 highschoolers, give them the SAT, give them the course, then give them the SAT again. Weโll treat both of these topics below. Remember the 6 steps: ๏ท State clearly what your variables are ๏ท Compute a test statistic ๏ท State the null and alternative hypotheses ๏ท Find the ๐-value ๏ท Decide upon a level of significance, ๐ผ ๏ท Form a conclusion
Looking at the picture it certainly seems like high bottom concentration values correspond to high top concentration values, and the same for low values. These data clearly come in pairs. A clever way to treat this case is by taking that fact seriously. Instead of seeing 12 data points, see 6 differences. Bottom Water Top Water Differences 0.430 0.415 0. 0.266 0.238 0. 0.567 0.390 0. 0.531 0.410 0. 0.707 0.605 0. 0.716 0.609 0.
If you want to test whether there is a difference on average between the bottom concentrations and the top concentrations, test whether the average difference is equal to zero. We know how to do this when the population of differences is normal, and that is what we will assume here.
๏ท State clearly what your variables are ๏ท Compute a test statistic ๏ท State the null and alternative hypotheses ๏ท Find the ๐-value ๏ท Decide upon a level of significance, ๐ผ ๏ท Form a conclusion
- State clearly what your variables are We will let ๐๐๐๐ก๐ก๐๐ represent the average zinc concentration on the bottom and similarly let ๐๐ก๐๐ represent the average zinc concentration on the top.
- State the null and alternative hypotheses ๐ป 0 : ๐๐๐๐ก๐ก๐๐ โ ๐๐ก๐๐ = 0 ๐ป 1 : ๐๐๐๐ก๐ก๐๐ โ ๐๐ก๐๐ = 0
- Decide upon a level of significance Take ๐ผ = 0.
- Compute a test statistic Since we have a small sample and we are testing means we hope to use the student t. Weโll assume that the population of differences is normally distributed (software can always help you to test this) and compute: ๐ก = ๐ฅ โ๐ ๐ ๐
= ๐ฅ๐๐๐ก๐ก๐๐^ โ ๐ฅ๐ก๐๐^ ๐ ๐โ^ ๐๐๐๐ก๐ก๐๐^ โ ๐๐ก๐๐
= ๐ฅ๐ ๐ ^ ๐โ^ ๐๐
Here we are saying that the โdifference of the averages ๐ฅ๐๐๐ก๐ก๐๐ โ ๐ฅ๐ก๐๐ works out to the same number as the average of the differences ๐ฅ๐. The same holds true for the population means. We will take ๐ ๐ to be the standard deviation of the differences and the sample size ๐ to be the number of differences, 6. Remember how to compute a standard deviation for a set of numbers. Weโll use the โcomputing formula: ๐ซ๐๐๐๐๐๐๐๐๐๐ ๐ซ๐๐๐๐๐๐๐๐๐๐๐ 0.015 0. 0.028 0. 0.177 0. 0.121 0. 0.102 0. 0.107 0. 0.55 0.
So that
๐ฅ๐ = 0.55 6 = 0.
๐๐^2 โ 1 ๐ ๐๐ ๐๐
๐ โ 1 =^
6 โ 1 =^
And then
๐ก = ๐ฅ โ๐ ๐ ๐
= 0.0916666670.060688275^ โ^0
Differences between Population Proportions It seems very natural to look at different populations and determine whether, for instance, high blood pressure is as common in men as it is in women. We already know how to handle tests in general. All we ever really need in a given situation is the proper statistic- everything else stays the same.
So, hereโs the statistic. If we were to repeatedly form samples of women of size ๐๐ค๐๐๐๐ , find out how many of them have high blood pressure, ๐๐ค๐๐๐๐ and divide to get
๐๐ค๐๐๐๐ = (^) ๐๐๐ค๐๐๐๐ ๐ค๐๐๐๐
And do the same for men
๐๐๐๐ = (^) ๐๐๐๐๐๐๐๐
And then considered the sampling distribution of the differences in the sample proportions (itโs a mouthful, but make sure you know what we are talking about) we would get roughly a normal distribution from the statistic
๐ง = (^) ๐๐๐ค๐๐๐๐^ โ ๐๐๐๐^ โ^ ๐๐ค๐๐๐๐^ โ ๐๐๐๐ ๐ค๐๐๐๐ 1 โ ๐๐ค๐๐๐๐ ๐๐ค๐๐๐๐ +^
as long as our sample sizes were decently large. Hereโs an example to make this a bit more concrete.
An experiment was conducted to test the effect of a new drug on a viral infection. The infection was induced in 100 mice after which the mice were randomly split into two groups of 50. The first group (called the treatment group) received the drug. The second group (called the control group) received no treatment for the infection. After a 30 day period, the proportions of survivors, ๐๐ก๐๐๐๐ก๐๐๐๐ก and ๐๐๐๐๐ก๐๐๐ , in the two groups were found to be ๐๐ก๐๐๐๐ก๐๐๐๐ก = 0.60 and ๐๐๐๐๐ก๐๐๐ = 0.36. Test at the ๐ผ = 0.05 level of significance whether the proportion of survivors was higher in the treatment group.
Note: Try this yourself before looking at the solution.
Six Steps:
- We will call the proportion of survivors in the hypothetical population of mice given the drug ๐๐ก๐๐๐๐ก๐๐๐๐ก and in the hypothetical population of mice not given the drug ๐๐๐๐๐ก๐๐๐. Note that we donโt have access to all the mice in the world so we donโt know what these numbers are. We just know them for our small sample.
- ๐ป 0 : ๐๐ก๐๐๐๐ก๐๐๐๐ก โ ๐๐๐๐๐ก๐๐๐ = 0 , ๐ป 1 : ๐๐ก๐๐๐๐ก๐๐๐๐ก โ ๐๐๐๐๐ก๐๐๐ > 0 (Do you see why this is a one tailed test?)
- ๐ผ = 0.
๐ง = (^) ๐๐๐ก๐๐๐๐ก๐๐๐๐ก^ โ ๐๐๐๐๐ก๐๐๐^ โ^ ๐๐ก๐๐๐๐ก๐๐๐๐ก^ โ ๐๐๐๐๐ก๐๐๐ ๐ก๐๐๐๐ก๐๐๐๐ก ๐ 1 โ ๐๐ก๐๐๐๐ก๐๐๐๐ก ๐ก๐๐๐๐ก๐๐๐๐ก +^
Note that your textbook (pooled estimate, page 464) makes a valid point here. If we are trying to see whether the data are consistent with ๐๐ก๐๐๐๐ก๐๐๐๐ก = ๐๐๐๐๐ก๐๐๐ = ๐ then we should construct our statistic as if this is true. Whatโs the best way to estimate ๐? We would do this by taking the number of surviving mice and dividing by the number of mice participating. That is, estimate this common value as
๐ = (^) ๐๐^1 +^ ๐^2 1 +^ ๐ 2
And then use
๐ง = ๐๐ก๐๐๐๐ก๐๐๐๐ก^ โ ๐๐๐๐๐ก๐๐๐^ โ^ ๐๐ก๐๐๐๐ก๐๐๐๐ก^ โ ๐๐๐๐๐ก๐๐๐ ๐ 1 โ ๐ ๐๐ก๐๐๐๐ก๐๐๐๐ก +^
๐ง = 0.6^ โ^ 0.36^ โ^0
