Sampling Distribution of Proportion - Hypothesis testing

Question

1. Suppose that 75% of apples are worm-free. Explain why it would not be unusual to find 80% or more of a simple random sample of 120 apples to be worm-free but it would be unusual to find 80% or more of a simple random sample of 480 apples to be worm-free. Use facts about the sampling distributions of sample proportions when n = 120 and when n = 480 to support your explanation.
2. The General Social Survey determined in 2016 that 57% of US Residents think that the use of marijuana should be made legal. In a 2018 survey of 1,578 US residents, 61% reported that they believed that the use of marijuana should be made legal. Do the 2018 data suggest a change in public opinion from 2016 to 2018? Do a hypothesis test at the 5% significance level.

Answer 1

The sampling distribution of a sample proportion is a probability distribution that shows the possible values of a sample proportion and how likely each value is. The sampling distribution of a sample proportion is approximately normal when the sample size is large enough.

In this case, we are told that 75% of apples are worm-free. This means that the mean of the sampling distribution of a sample proportion of 120 apples is 0.75. The standard deviation of the sampling distribution is 0.071.

The probability of finding 80% or more of a simple random sample of 120 apples to be worm-free is very small. This is because the standard deviation of the sampling distribution is relatively small, so the sample proportion is likely to be close to the population proportion.

In contrast, the probability of finding 80% or more of a simple random sample of 480 apples to be worm-free is much larger. This is because the standard deviation of the sampling distribution is much smaller, so the sample proportion is more likely to be further from the population proportion.

The General Social Survey determined in 2016 that 57% of US Residents think that the use of marijuana should be made legal. In a 2018 survey of 1,578 US residents, 61% reported that they believed that the use of marijuana should be made legal.

To do a hypothesis test at the 5% significance level, we would set up the following hypotheses:

H0: The proportion of US residents who think that the use of marijuana should be made legal has not changed from 2016 to 2018.

Ha: The proportion of US residents who think that the use of marijuana should be made legal has increased from 2016 to 2018.

We can calculate the test statistic as follows:

z = (p_hat - p_0) / sigma_p

where:

z is the test statistic

p_hat is the sample proportion of 61%

p_0 is the hypothesized population proportion of 57%

sigma_p is the standard error of the sampling distribution, which is 0.019

Plugging in these values, we get a test statistic of 2.05.

The critical value for a two-tailed test at the 5% significance level is 1.96. Since our test statistic is greater than the critical value, we can reject the null hypothesis and conclude that there is enough evidence to support the alternative hypothesis.

In other words, the 2018 data suggest that public opinion on the use of marijuana has changed from 2016 to 2018.