A recent random sample of $\mathrm{n}=805$adult U.S. residents found that the proportion who rated the honesty and ethical standards of nurses as high or very high is $0.85$. This is $0.15$higher than the proportion recorded for doctors, the next highest-ranked profession.

a. Identify the sample and the population in this setting.

b. Do you think that the proportion of all U.S. residents who would rate the honesty and ethical standards of nurses as high or very high is exactly $0.85$? Explain your answer.

c. What is the benefit of increasing the sample size in this context?

We need to identify the sample and the population in the setting.

The population contains all individual subjects for which we get information.

A sample is the part of the population from which information was actually collected.

We then note that the sample is the $805$adult U.S. residents in the random sample, then the population then needs to be all adult U.S. residents.

So,

$$\begin{gathered} \mathrm{population}=\mathrm{All}\mathrm{adult}\mathrm{U}.\mathrm{S}.\mathrm{residents} \\ \mathrm{Sample}=805\mathrm{adult}\mathrm{U}.\mathrm{S}.\mathrm{residents}\mathrm{in}\mathrm{the}\mathrm{random}\mathrm{sample} \end{gathered}$$

We need to find the proportion of all U.S. residents who would rate the honesty and ethical standards of nurses as high or very high is exactly$0.85$or not.

The population contains all individual subjects for which we get information.

A sample is the part of the population from which information was actually collected.

We then note that the sample is the $805$adult U.S. residents in the random sample, then the population then needs to be all adult U.S. residents.

We then do not expect the same proportion to be exactly equal to the population proportion, because we expect some sampling variability which implies that the estimates in the samples are expected to vary slightly from the actual value in the population.

So, the answer is no.

We need to find the benefit of increasing the sample size in this context.

If the increase the sample size is, then the sample will contain more information about the population, and the estimates of the population are then expected to be more accurate when using a large sample to make the estimates.

This implies that the sample values will be closer to the corresponding population values.