1. Sickle-cell and malaria: Sickle-cell anemia is hereditary condition that is common among African-Americans and can cause medical problems. Some biologists suggest that the sickle-cell trait protects against malaria. That would explain why it is found in people whose ancestors originally came from Africa, where malaria is common. A study in Africa tested 523 children for the sickle-cell trait and also for malaria. Here are the results.

Severe malaria infection

Yes    No

Sickle-cell trait    31    95

No sickle-cell trait    147    250

a. Estimate the proportion of severe malaria infection in those with the sickle-cell trait and those without it. In each case, provide the 95% confidence interval (CI) and interpret its value..

b. Based on the estimates and their confidence intervals, what does this tell you about the possible relationship between malaria infection and sickle-cell?.

c. Now we wish to verify whether there is an association between malaria and sickle cell using a chi-square test.

a. What are the null and alternative hypotheses for this test?

b. Find the expected cell counts under the null hypothesis. Verify that the expected counts meet the guideline for use of the chi-square test.

c. Compute the chi-square statistic and find p-value for this test.

d. What is your conclusion?  Is there a good evidence of a relationship between the sickle-cell trait and severe malaria infection?  Why or why not?

d. An alternative approach to evaluating the association between the outcome (severe malarial infection) and exposure (sickle-cell trait) is to test the difference of the risks (proportions) using a z-score.

a. What are the null and alternative hypotheses of this test if it was a two-tailed test?

b. What is the point estimate and its standard error?

c. Compute the test statistic and determine whether there is a significant difference in proportions using an alpha of 5%. State your conclusions and discuss how they compare to the conclusions in both (b) and (c).