Question

In the US I want to know what proportion of 13-year-olds vape. Three distributions are associated with this question: population distribution, sample distribution, and sampling distribution. Describe the role of each in answering this question.

In cross-sectional analysis explain why it is necessary to take a random sample.

Describe the difference between an estimator and an estimate and explain why estimators are random variables while estimates are not.

Explain why point estimates are always wrong.

Answer 1

1)Population distribution

The population is the whole set of values, or individuals, you are interested in. Population characteristic are mean (μ), Standard deviation (σ) , proportion (P) , median, percentiles etc. The value of a population characteristic is fixed. This characteristics are called population distribution. They are symbolized by Greek characters as they are population parameters2) Sample Distribution

The sample is a subset of the population, and is the set of values you actually use in your estimation. This sample has some quantity computed from values e.g. mean (x ), Standard deviation (s) , sample proportion etc. This is called sample distribution. The mean and standard deviation are symbolized by Roman characters as they are sample statistics.

3)Sampling Distribution

Researchers often use a sample to draw inferences about the population that sample is from. To do that, they make use of a probability distribution that is very important in the world of statistics: the sampling distribution. It is theoretical distribution. The distribution of sample statistics is called sampling distribution.

In context to US population we can take a total population of 13 years old who vape and from that population we can take a few samples which will represent the population in best possible way and for each of the sample value we will calculate the statistic and combine distribution of these statistic will represent the sampling distribution.

For determining the proportion of 13 years who vape will be divided by the total population of 13 years age group.

A c​​ross sectional study is defined as an observational research type that analyzes data of variables collected at one given point of time across a sample population. population or a pre defined subset .Basically in cross sectional study we analyze different factors of a population for a same point of time.It is important to take a random sample because we can't analyze the whole population for a point of time with respect to various factors we can only study a part of it.

An estimator is a function that maps samples into your parameter space. An estimate is the value of that function taken on a particular sample. An estimator is a statistic that you apply to data in order to obtain the estimate.Being a function of the data, the estimator is itself a random variable; a particular realization of this random variable is called the estimate.

Point estimators are functions that are used to find an approximate value of a population parameter from random samples of the population. They use the sample data of a population to calculate a point estimate or a statistic that serves as the best estimate of an unknown parameter of a population. It is always wrong because we can not estimate accurately the population parameter from the sample data.There will always be some discrepancy between the actual value and the estimated value so there is presence of some error term which is actually the difference between the actual value and the estimated value.