Chapter 9 notes
The video above is from a past semester. The material is the same but any dates mentioned will be incorrect. For the current term, please refer to the due dates in syllabus.
This chapter is very similar to chapter 8. The difference being that this chapter has two samples rather than one. Same theory applies but the test stats are more complicated. If you use Statcrunch, however, there is very little difference as Statcrunch does all the heavy lifting.
Section 9-1 Inferences about two proportions
Assumptions:
1) Two SRS that are independent
2) At least 5 successes and 5 failures for each of the two samples
The test stat and confidence interval calculations for this section are intense looking but the theory is exactly the same as we've seen in the last two chapters . I will work out several of these below to show you how they work but please don't calculate all this by hand, use STATCRUNCH!!
Video working a two proportion hypothesis test and confidence interval on Statcrunch (example of the above and below type problems)
Below is the confidence interval for the bednet problem. Note that the significance level is 0.01 yet I use a 98% confidence level. This is due to the hypothesis test being one tailed. When we have a one tailed hypothesis test the full alpha goes in the single tail but when we switch to a confidence interval we need that value of alpha in both tails (in this case 0.01 in each tail) reducing the confidence down to 98% (1-0.01-0.01). This is only true for one tailed tests and not two tailed.
Section 9-2 Inferences About Two Means (Independent Samples)
Assumptions:
1) Population standard deviations are unknown, and no assumption is made about the equality of them.
2) Independent samples
3) SRS
4) n1>30 and n2>30 and/or both normally distributed
Do different species of fish have different levels of Methyl-Mercury? My friends and I catch various different species of fish off the coast of California and preformed a scientific study to determine the levels of Methyl-Mercury in these species. Below is a subset of our data from two of those species: the Quillback and the Canary rockfish. The Quillback is a very slow growing fish and the Canary is much faster growing, both grow to comparable sizes. Our assumption was that the Quillback would have higher Methyl-Mercury levels due to being and older fish. Listed below are the Methyl-Mercury levels from each of the species samples. Use a 0.05 significance level to test the claim that Quillback rockfish have higher Methyl-Mercury levels than Canary rockfish. (Samples are small, assumption is made that they come from a population that is normally distributed)
Quillback Methyl-Mercury Levels
1.04
0.747
1.4
1.41
1.38
0.128
0.853
1.05
0.286
Canary rockfish Methyl-Mercury Levels
0.087
0.175
0.109
0.321
0.213
0.177
0.219
0.218
0.62
SOLUTION: Since we have raw data, we preform the calculation in the following manner:Copy the data into Statcrunch, click Stat, T Stats, 2-sample, with data. Select the two data sets and enter in the Ho and Ha (remember this is a right tailed test since our claim is that Quillback have higher Methyl-Mercury than Canary). Click compute.
Two sample T hypothesis test:
μ1 : Mean of Quillback Methyl-Mercury Levels
μ2 : Mean of Canary rockfish Methyl-Mercury Levels
μ1 - μ2 : Difference between two means
H0 : μ1 - μ2 = 0
HA : μ1 - μ2 > 0
(without pooled variances)
Hypothesis test results:
Difference Sample Diff. Std. Err. DF T-Stat P-value
μ1 - μ2 0.68388889 0.16576197 9.7897728 4.1257285 0.0011
Initial conclusion: Since the p-value(.0011) < alpha (0.05), we reject Ho. Final conclusion: There is sufficient evidence to support the claim that Quillback rockfish have higher Methyl-Mercury levels than Canary rockfish. Our assumption appears to be correct, older/slower growing fish accumulate higher levels of Methyl-Mercury.
Section 9-3 Two-Dependent Samples (Matched Pairs)
Assumptions:
Matched Pairs
SRS
n>30 or the pairs of values have differences that are from a population with a normal distribution.
9-3 # 8 Listed below are the number of words spoken in a day by each member of eight different couples. The data are randomly selected from the first two columns in data set 24 “Word counts” in Appendix B.
Use a 0.05 significance level to test the claim that among couples, males speak fewer words in a day than females.
Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a).
male 15,684 26,429 1,411 7,771 18,876 15,477 14,069 25,835
female 24,625 13,397 18,338 17,791 12,964 16,937 16,255 18,667
d -8941 13032 -16927 -10020 5912 -1460 -2186 7168
d^2 79941481 169833024 286523329 100400400 34951744 2131600 4778596 51380224
The d and d^2 are calculated above to find the Sd (standard deviation of the differences). Again, I recommend using statcrunch for these as the calculations are lengthy.
