Licensed vaccines

QUESTION 1

With no currently licensed vaccines to inhibit malaria, good news was welcomed with a recent study reporting long-awaited vaccine success for children in Burkina Faso. With 450 children randomized to either one of two different doses of the malaria vaccine or a control vaccine, 89 of 292 malaria vaccine and 106 out of 147 control vaccine children contracted malaria within 12 months after the treatment.
A randomization test was conducted; the histogram of simulated differences under the null hypothesis is shown below.

Consider the hypothesis test constructed to show a lower proportion of children contracting malaria on the malaria vaccine as compared to the control vaccine. Write out the null and alternative hypotheses, estimate a p-value using the randomization histogram, and conclude the test in the context of the problem.

QUESTION 2
For the malaria vaccine effectiveness study, we generate 1,000 boostrapped differences in the proportion of children who contracted malaria with the malaria vaccine versus the control vaccine (p_malaria - p_control). These are shown in the histogram below.

A. What is the lower bound of the 95 percent bootstrap confidence interval? Enter your answer as a decimal, between -1 and 1; your answer may have approximation error, up to 0.025 in absolute magnitude.

B. For the same bootstrap distribution of the difference in proportions between the malaria vaccine group and the control vaccine group, what is the upper bound of the 95 percent bootstrap confidence interval.

QUESTION 3

1. Researchers studying a community of Antarctic penguins collected body measurement (bill length, bill depth, and flipper length measured in millimeters and body mass, measured in grams), species (Adelie, Chinstrap, or Gentoo), and sex (female or male) data on 344 penguins living on three islands (Torgersen, Biscoe, and Dream) in the Palmer Archipelago, Antarctica. The summary table below shows the results of a linear regression model for predicting body mass (which is more difficult to measure) from the other variables in the dataset.
   term estimate std.error statistic p.value
   (Intercept) -1461.0 571.3 -2.6 0.011
   bill_length_mm 18.2 7.1 2.6 0.0109
   bill_depth_mm 67.2 19.7 3.4 7e-04
   flipper_length_mm 15.9 2.9 5.5 <0.0001
   sexmale 389.9 47.8 8.1 <0.0001
   speciesChinstrap -251.5 81.1 -3.1 0.0021
   speciesGentoo 1014.6 129.6 7.8 <0.0001
   Single choice question
2. Write the equation of the regression model.
   Predicted body_mass = -1461 + 18.2bill_length_mm + 67.2bill_depth_mm + 15.9flipper_length_mm + 389.9sexmale - 251.5speciesChinstrap + 1014.6speciesGentoo

Predicted body_mass = 18.2bill_length_mm + 67.2bill_depth_mm + 15.9flipper_length_mm + 389.9sexmale - 251.5speciesChinstrap + 1014.6speciesGentoo

Observed body_mass = -1461 + 18.2bill_length_mm + 67.2bill_depth_mm + 15.9flipper_length_mm + 389.9sexmale - 251.5speciesChinstrap + 1014.6speciesGentoo

Predicted body_mass = -1461 + 18.2bill_length_mm + 67.2bill_depth_mm + 15.9flipper_length_mm + 389.9sexmale - 251.5speciesChinstrap + 1014.6speciesGentoo + residual

Observed body_mass = the actual data point

2. Interpret the slope of flipper length (in millimeters).
   All else held constant, for each additional millimeter flipper length is higher, body mass of penguins is expected to be higher, on average, by 15.9 grams, but this is only true for male penguins.

All else held constant, for each additional millimeter flipper length is higher, body mass of penguins is expected to be higher, on average, by 15.9 grams.

All else held constant, for each additional gram body mass is higher, flipper length of penguins is expected to be higher, on average, by 15.9 millimeters.

For each additional millimeter flipper length is higher, body mass of penguins is expected to be higher, on average, by 15.9 grams.