Biometry Assignment #4 -Inference for a Single Categorical Variable
Spring 2017- Due Tuesday, February 21st by Midnight
For questions where you asked to conduct a test, do the following:
- Identify the parameter(s) of interest (Only for problems 1, 2, & 3)
- State the null and alternative/research hypothesis
- Conduct an appropriate test obtaining a p-value that can be used to draw a conclusion regarding the hypotheses
- Finally state your conclusion in practical terms, i.e. do not just say “Reject Ho”.
For questions where you are simply asked to find a confidence interval, just find the interval you are asked to find and give an appropriate summary of the confidence interval, e.g. “We estimate that the proportion of animals with the trait defect is…”.
Feel free to use JMP if you want. If you do use JMP be sure to copy and paste all relevant results into your homework.
1. Proportion of Animals with Trait Defect
It is believed that 25% of a certain animal will exhibit a certain trait defect. In a
random sample of 1215 animals, 362 exhibited a certain trait defect.
Research Question:Do these data suggest that the percentage of animals with this trait is different than 25%?
a) Conduct an appropriate test to answer this question. (5 pts.)
b) Determine the 95% confidence interval for the proportion of the population displaying
the trait. Also give the correct interpretation of this interval. (3 pts.)
c) Does the CI from part (b) support your conclusion from part (a)? Explain. (2 pts.)
2. A severe drought in 1987 affected both the death rate of seedlings and the growth rate of established trees. It is thought that a majority of the trees in the affected areas have a 1987 growth ring that is less than one half the size of the trees' other growth rings. A sample of 250 trees yielded 150 with this characteristic.
Research Question:Do these data support the claim that a majority of trees exhibited the growth defect?
a)What is the parameter of interest in this study? (1 pt.)
b)Conduct a test answer the research question. (5 pts.)
c)Construct and interpret a 95% CI for the parameter of interest in this study. (3 pts.)
4. Carpal Tunnel Syndrome can be treated both surgically and with through the use of splints. In study comparing the effectiveness of each 176 patients were randomly assigned to two group of 88 subjects. One group of subjects had a surgery to alleviate symptoms associated with Carpal Tunnel Syndrome, the other group were treated with splints. In the surgery group 71 of the 88 patients showed an improvement in their symptoms and in the splint group 47 of the 88 patients showed an improvement. The results are summarized in the table below.
(Note: this is very similar to barbless vs. barbed hook example done in class)
Surgery / 71 / 17 / 88
Splints / 47 / 41 / 88
Column Total / 118 / 58 / n = 176
a)Estimate P(Improvement|Surgery) (1 pt.)
b)What proportion of patients in the surgery group showed improvement? (1 pt.)
c)Estimate P(Improvement|Splint) (1 pt.)
d)What proportion of patients in the splint group showed improvement? (1 pt.)
e)Construct a 95% CI for the proportion of Carpal Tunnel Syndrome patients who will show improvement when surgery is used, i.e. find a 95% CI for (3 pts.)
f)Construct a 95% CI for the proportion of Carpal Tunnel Syndrome patients who will show improvement when splints are used, i.e. find a 95% CI for (3 pts.)
g)Do these confidence intervals suggest that in the population of Carpal Tunnel Syndrome patients that surgery is more effective
h)What is the Relative Risk (RR) for improvement, or better stated the “Relative Benefit” associated with surgery? Interpret. (2 pts.)
i)What is the Odds Ratio (OR) for improvement associated with surgery? Interpret. (2 pts.)
Note: You should notice that the OR is much larger than the RR. This happens when the probability of the event in question is large. In situations where you have the option of using the RR or the OR to summarize results like these, you should always use the RR if the probability of the event is large!
5. Heron Island is a coral cay in the Great Barrier Reef. Around Heron Island (and elsewhere in Australia) there are 2 color morphs of the reef heron. Egretta sacra, a white morph and a dark or blue morph. It is generally accepted that further north there are many more white than dark morphs, while further south just the opposite is true. A preliminary study was carried out to test the hypothesis that the ratio of white to dark herons on the island was 3:1.
a) A small sample of 20 herons, found 16 white morphs and 4 dark morphs. Can the assumption of 3:1 ratio be rejected? (3 pts.)
b) What if a large sample of 200 herons was taken and it yielded 158 white morphs and 42 dark morphs, can the assumption of 3:1 ratio be rejected? Note: as there are only two categories you should NOT use a chi-square goodness of fit test. (3 pts.)
6. DeMeis & Stearns in their study of age effect on academic and social performance recorded the month of birth. In their sample of n = 2,802 students they observed the following distribution of births across the 12 months.
Month / Jan / Feb / Mar / Apr / May / Jun / Jul / Aug / Sep / Oct / Nov / DecBirths / 258 / 195 / 216 / 251 / 242 / 267 / 234 / 225 / 215 / 257 / 228 / 214
Expected
Frequency
a)Are the expected numbers of births for each month if we assume that births are uniformly distributed across the months of the year? Put these expected frequencies in the cell provided. (2 pts.)
b)Is there evidence that the births are not uniformly distributed across months?
(4 pts.)