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

Answer 1

The smallest sample size to satisfy the conditions is n=500.

Explanation:

The condition for performing the inference related to the sample size is that, for all the categories, the expected success and failures in the sample are at least 10:

`np\geq10\\\\n(1-p)\geq10`

The largest sample size required will be for the minimum p or (1-p).

This happens to be the proportion that can play other instruments: 2%.

Then, we can calculate the minimum sample size as:

`np\geq10\\\\n(0.02)\geq10\\\\n\geq(10/0.02)\\\\n\geq500`

Answer 2

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.