Question

It is believed that 43% of the US population can play the piano, 28% can play the guitar, 15% can play the harmonica, 12% can play the drums, and 2% can play other instruments. You want to take a simple random sample of individuals to test this claim. What is the smallest number of people required for the sample to meet the conditions for performing inference?

250
150
100
43
2

Answer 1

250

Explanation:

Remember there are 2 conditions to perform a goodness of fit chi-test:

Simple random sample: The data must come from a random sample or a randomized experiment.

Expected counts: All expected counts are at least five. You must state the expected counts.

To explain expected counts a bit better, imagine I surveyed 100 people about their ice cream preferences. Before beginning it is believed that 50% like chocolate, 47% like vanilla, and 3% like strawberry.

That means our expected counts are:

100(.50) = 50

100(.47) = 47

100(.03) = 3

This is a problem, because 3 < 5 and so we can not perform a goodness of fit chi-test.

So how do you find the minimum sample size? Use this formula:

sample size (n) * smallest proportion (p) = 5

In the context of ice cream:

n*.03 = 5

n = 5 / .03

n = 167 (because you can't interview 2/3s of a person)

In the context of the problem:

n* .02 = 5

n = 5 / 0.02

n = 250

This means we need to sample at least 250 people to meet our expected count condition.

Answer 2

43% = 'Piano'

28% = 'Guitar'

15% = 'Harmonica'

12% = 'Drums'

2% = ...Other Instruments

- '2' Does seems as to be the smallest but it could be a trick.

- '43' Is the largest percentage out of the whole group.

- '100' Is what you get when you add all of the percentages together.

- And if you try to evaluate the percentages together in either way it'll end up to be '0.00004334' so that executes '250' and '150' from out of the question.

I'm not sure exactly what the actual answer could be but I'm assuming since it said "What is the smallest number of people required for the sample to meet the conditions for performing inference" Then my assumption is '2'.