According to a poll, 20% of adults smoke. In a random sample of 75 adults, 30 are smokers. Is this unusual?

yeah i guess

To determine whether this is unusual or not, we can compare the observed proportion of smokers in the random sample to the expected proportion based on the poll.

First, let's calculate the expected number of smokers in the sample. We know that 20% of adults smoke, so we can find the expected proportion of smokers in the sample by multiplying 20% by the sample size:

Expected Proportion of Smokers = 20% = 0.2
Expected Number of Smokers = Expected Proportion of Smokers × Sample Size
= 0.2 × 75
= 15

Now, let's compare the expected number of smokers to the observed number of smokers in the sample.

Observed Number of Smokers = 30

Since the observed number of smokers (30) is greater than the expected number of smokers (15), it suggests that the proportion of smokers in the random sample is higher than the overall population proportion.

To determine if this difference is statistically significant, we need to perform a hypothesis test. One commonly used method is the proportion test, specifically the one-sample z-test for proportions.

The steps for conducting a one-sample z-test for proportions are as follows:

1. Formulate the null hypothesis (H0) and the alternative hypothesis (H1):
- H0: The proportion of smokers in the random sample is equal to the proportion of smokers in the overall population (p = 0.2).
- H1: The proportion of smokers in the random sample is different from the proportion of smokers in the overall population (p ≠ 0.2).

2. Calculate the test statistic, which in this case is the z-score:
z = (Observed Number of Smokers - Expected Number of Smokers) / sqrt(Expected Number of Smokers × (1 - Expected Number of Smokers) / Sample Size)

3. Determine the critical value based on the desired level of significance (α). This value is compared to the absolute value of the test statistic to determine statistical significance. For example, if α = 0.05, we would compare the absolute value of the test statistic to the critical value of 1.96 (for a two-tailed test).

4. Compare the absolute value of the test statistic to the critical value:
- If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis and conclude that the difference is statistically significant.
- If the absolute value of the test statistic is smaller than or equal to the critical value, we fail to reject the null hypothesis and conclude that the difference is not statistically significant.

By following these steps and performing the necessary calculations, we can determine if the observed proportion of smokers in the random sample is significantly different from the proportion obtained from the poll.