Question

Seat belt use in WA state (pop: 7.5 million) is reported to be 95% compared to 85% in Mississippi (pop 3 million). A random sample of 1,000 residents is taken from each state. Since we are sampling a higher proportion of the population of Mississippi, the % of seatbelt users in the Mississippi sample will likely be closer to the population proportion. Be sure to state true or false clearly first. Then explain which sample (WA or MI) is more likely to be closer to its population proportion and why?

Answer 1

True

Explanation:

By sampling a larger proportion of Mississippi's population, the percentage of people who wear seatbelts in Mississippi surveys would be closer and nearer to the population proportion.

The rule of large numbers states that if we collect samples of increasing size from every population (here, Mississippi vs. Washington WA state), the sampling distribution's mean, px, continues to get closer and nearer to the actual true population proportion (p).

Recall that as n becomes greater, the sample mean obeys a normal distribution because of the Central Limit Theorem. However, as (n) rises, the sampling distribution's standard deviation reduces.

where:

sampling proportion's standard deviation =
`\sqrt{(p((1-p))/(n)}`

If n rises, the sample proportion
`P_x` have to get nearer to the population proportion P.

For both counties, the sample size is 1000, but the true population varies. According to the provided statement, the true population of Mississippi (3 million) is less than that of Washington State (7.5 Million). As a result, we can confidently assume that our sampling proportion in Mississippi seems to be much closer to the true actual population than our sampling proportion in Washington state.