The following problems are copied from the chapter 17 exercises from Introduction to Modern Statistics First Edition by Mine Çetinkaya-Rundel and Johanna Hardin (https://openintro-ims.netlify.app/inference-one-prop.html)

The following problem is a modified version of the problem that appears in your book.

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. (Datoo et al. 2021)

  1. Is this data collected in an experiment or an observational study? How do you know?

ANSWER: Experiment, as the children were assigned a treatment or control group by random assignment.

I created a data set called malaria_data for you to use when answering the following questions.

##                  outcome
## treatment         infected not infected Sum
##   control vaccine      106           41 147
##   malaria vaccine       89          203 292
##   Sum                  195          244 439
  1. In both words and symbols provide the parameter(s) of interest for this study.

ANSWER: \(p_m\) is the proportion of patients that were infected when given the malaria vaccine \(p_c\) is the proportion of patients that were infected when given the control vaccine

  1. Construct a plot to display the proportion of children that were infected with malaria within each treatment.
malaria_data %>% 
  select(treatment, outcome) %>% 
  table(dnn=c("malaria vaccine", "control vaccine")) %>%
  prop.table (1) %>%
  addmargins()
##                  control vaccine
## malaria vaccine    infected not infected       Sum
##   control vaccine 0.7210884    0.2789116 1.0000000
##   malaria vaccine 0.3047945    0.6952055 1.0000000
##   Sum             1.0258830    0.9741170 2.0000000
  1. Calculate the statistics that correspond to the parameters you identified in part b.

ANSWER: \(\hat p_m =\) .304

\(\hat p_c =\) .721

  1. 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

ANSWER: \(H_0p_m = p_c\) The proportion of patients that got infected that received the malaria vaccine and control vaccine are not significantly different

\(H_a p_m< p_c\) The proportion of patients that got infected that received the malaria vaccine was significantly less than the proportion that received the control vaccine.

  1. Estimate the p-value using the randomization histogram below.

ANSWER:

  1. State the conclusion of the test in the context of the problem. Use a significance level of .05.

ANSWER: