# mathematics

1. [Learning process check-up] This week we continue the topic on hypothesis test of one categorical variable (sample) and start the topic on investigating the relationship between two categorical variables (samples).

2. Primary lecture recording: Topic 11.2 and 12.1

[Review question 1] Under a genetic model of dominant epistasis, a cross of white and yellow summer squash will yield white, yellow, and green squash with probabilities 12/16, 3/16 and 1/16 respectively. The observed data are given in the table.

Perform the following non-directional hypothesis test:

a) Ho: Pr(white)=12/16 versus Ha: Pr(white)

Then perform the following directional hypothesis test:

b) Ho: Pr(white)=12/16 versus Ha: Pr(white) >12/16

Last, the other directional hypothesis test:

c) Ho: Pr(white)=12/16 versus Ha: Pr(white) <12/16

StatsPortal> assignment> StatTutor: The Chi-square test for goodness of fit

[Review question 2] Can we use a chi-square goodness of fit test to answer the following question? ________

“Do the colors of M&M’s in a 16 ounce bag follow the proportions of colors posted on the M&M website?”

Which of the following statements about the data set is correct?

A. There are two samples of M&M’s with three or more colors.

B. There are two types of M&M’s with three or more colors.

C. It is one sample of M&M’s with more than two colors.

[Review question 3] How should data be collected for a chi-square goodness-of-fit test? ________

A. One simple random sample (SRS)

B. Separated simple random samples, one from each population

C. Randomized order of applying two treatments to each individual in a matched pair design

D. Random allocation of individuals to two or more treatments

[Review question 4]: For comparison of two population proportions, p1-p2, under what conditions could you assume that the difference follows a Normal distribution? What is its mean and standard deviation?

[Review question 5]: Refer to the example of ecological approach to disease eradication that covered in Topic 12.1. Lyme disease is spread by infected ticks. Ticks feed mainly on mice. Mice feed on acorn. An experiment compared two similar forest areas in a year with low acorn amounts. One area was supplied large amounts of acorns, and the other untouched. The next spring mice populations were compared.

We want to compare the true (population) proportions of breeders found in the two areas, . Compute and interpret a 95% confidence interval for .

1. Study of staph infections in surgery patients: The researchers recruited 917 patients who had tested positive for Staphylococcus aureus and randomly assigned them to receive a nasal ointment that contained either a staph-killing solution or a placebo. Postsurgery infections with S. aureus were then recorded. In the treatment group, 17 of 504 patients developed a staph infection, compared with 32 of 413 patients in the placebo group. Calculate and interpret a 95% confidence interval for the difference between the proportions of postsurgical staph infections among patients receiving a staph-killing ointment or a placebo.

2. Resource-selection analysis compares the distributions of animals relative to the distribution of habitat. If the two don’t agree, there is evidence of selection. A survey of 106 moose found that 24 were located in “In burn – interior”, 22 in “In burn – edge”, 20 in “Out of burn – edge”, and 40 in “Other” habitat types. The corresponding proportion of the landscape as determined by a Geographic Information System (GIS) in these habitats was 0.340, 0.101, 0.104, and 0.455 respectively.

a) What is the null and alternative hypotheses?

b) Complete the chi-square GOF table:

Categories In burn-interior In burn-edge Out of burn other observation(expectation)

c) Compute the chi-square statistic.

d) Find the p-value of the test and draw a conclusion.

3. Refer to Question 2, conduct the following directional test: the proportion of “other” is lower than 45.5%.

a) Define the hypothesis.

b) Complete the chi-square GOF table.

Categories “other” “non-other” observation(expectation)

c) (A unique procedure for directional test only) Check whether the sample proportion is consistent with Ha. That is, if Ha states P is less than a value, then the sample proportion is also less than this value.

d) Compute the chi-square statistic.

e) Find the p-value and draw a conclusion.

5. [OQ]. Frist complete the following question, then create a similar question

Consider a fictitious population of mice. Each animal’s coat is either black (B) or grey (G) in color and is either wavy (W) or smooth (S) in texture. Express each of the following relationships in terms of probabilities or conditional probabilities relating to the population of animals.

For example: “Smooth coats are more common among black mice than (smooth coats are) among grey mice” is presented as Pr(S | B)> Pr(S |G)

a) Black mice are more smooth than wavy.

b) Grey mice are more likely to have wavy coats than black mice.

c) Smooth coats are more often black than are wavy coats.

d) Smooth coats are more often black than grey.

e) Smooth coats are more common than wavy coats.