**Constructing a 95% Confidence Interval for Population Mean Income**To estimate the population mean annual income for individuals over age 40 in your state, a sample is chosen. The sample's mean income is calculated as $40,495, with a standard deviation of $7,782.**(a) Best Point Estimate:**What is the best point estimate for the population mean, based on the sample?- **Answer:** $40,495**(b) Determining Distribution for Confidence Interval Calculation:**For different sampling scenarios, determine which distribution—Z (standard normal distribution) or t (t-distribution)—should be used to calculate the critical value for the 95% confidence interval for the population mean.- **Sampling Scenarios:** - **Scenario 1:** - Sample size: 12 - Population: Normally distributed with an unknown standard deviation. - **Distribution to Use:** t - **Scenario 2:** - Sample size: 85 - Population: Non-normally distributed with a known standard deviation of 7,613. - **Distribution to Use:** Z - **Scenario 3:** - Sample size: 19 - Population: Distribution with little known information. - **Distribution to Use:** UnclearThis schematic helps identify the appropriate statistical methods for constructing confidence intervals in various sampling contexts, emphasizing the criteria for using Z or t-distributions based on sample size and population characteristics.

Given that sample mean = 40,495 dollars Sample standard deviation = 7782 dollars(a) From the theory of estimation, it is known that sample mean is a consistent estimator of the population mean. Therefore, we can use the sample mean as a best point estimate for population mean The answer is 40,495 dollars.(b) When sample size n = 12 < 30 a small sample, population is normally distributed and population standard deviation is unknown which are the the assumption required to use t-distribution. Therefore, we use t-distribution When the sample size n = 85 > 30 a large sample, then by central limit theorem we have the sample mean of any distribution approximately follows normal distribution. Therefore, we use Z or normal distribution. When the sample size n = 19 < 30 a small sample and a little information about the population is known then there are two situations. If the population comes out to be normal, use t-distribution If the population comes out to be non-normal, use non parametric test. Therefore, the answer is unclear for the question.