Worksheet 16: Sample Size Matters#

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1. Model the hot dog poll with either a bag of marbles or with coinflips.

2. Are we guaranteed that the sample mean $\widehat{{\mu }_n}=\mu$, the population mean? Why?

3. In the microplastics experiment, can we directly compute the probability that $|\widehat{{\mu }_n}-\mu |$ is big?

4. What do you expect to happen to our estimate $\widehat{{\mu }_n}$ as $n$ gets larger?

5. As we increase $n$, what do you notice about:

   a. The variability of the dataset of our estimates ${\hat{\mu }}_n$?

   b. The shape of the histogram?

6. Formulate the following as a “population vs. sample” scenario. a. What is the population? b. What is the variable $x$ that describes members of the population? What is the population mean $\mu$? c. What are our samples $x_1,\dots ,x_n$?

   A medical researcher has come up with a new drug. The drug has some side effects; a headache that lasts anywhere from $0$ to $48$ hours.

   The researcher designs an experiment to determine the average duration of the side-effect headache; they recruit a group of $n$ random sick patients, gives them all the drug, records the length of each of their headaches, and calculates the mean.