Quiz-4

Short answer solutions

Question:

Suppose we wanted to test if a coin was fair. We start by assuming that the coin is fair, and want to investigate if it lands on heads more than 50% of the time.

Assume the null value is .50.

Assume that your sample statistic was 6 heads and 44 tails.

Do you expect the null distribution to be approximately normal? Why or why not? For this question, you can assume independence between coin flips.

Solution: Because we are testing this condition under the assumption that the true proportion of heads is equal to 50%, we need to check this condition under this assumption. In a sample size of 50 (n), how many successes and how many failures would we expect?

50*.5 = 25 successes

50*.5 = 25 failures

Both of these values are larger than 10, so we would expect the sampling distribution under the assumption of the null hypothesis to be normally distributed.

Question:

Suppose you want to test if more than 50% of female penguins are on an island. In your random sample of penguins, you observe 35 female penguins and 15 male penguins.

You may assume that your null value is equal to 50%.

In a simulation study, how would one dot on the simulated sampling distribution be created?

Solution:

To approximate our sampling distribution under the assumption of the null hypothesis, we need to simulate some data first. We need to use a random process that can mimic the assumption that the true proportion of female penguins is really equal to 50%. We could use a coin, or we could use a spinner, or we could use cards. I’m going to use a coin.

Let heads represent a female penguin, and tails represent a male penguin. We can then flip a coin 50 times, and calculate the new simulated sample proportion of female penguins over 50. This new proportion is one dot, or observation, on our sample sampling distribution under the assumption of the null hypothesis.