Complete the following TWO essays using examples and opinions in your OWN words.

ID: 452903 • Letter: C

Question

Complete the following TWO essays using examples and opinions in your OWN words. Use your OWN words beyond definitions and textbook. As a guide each essay should be about one side of one page in length.

A. Chapter 8 - Sampling Distributions and Estimations:

Two types of confidence intervals for a mean were discussed one with known sigma and one with sigma unknown. Give and discuss a real example of how this would be used and describe the different test statistics that are used.

B. Chapter 9 - One-Sample Hypothesis Tests:

Hypothesis testing along with type I and Type II errors errors are important in setting up a scenario in the real word to accept or reject the null hypothesis. Give a real word example outside of the text of how you would set up a hypothesis test and decide whether to accept or reject the null hypothesis. Also briefly discuss the importance of the two type errors

Explanation / Answer

Plot the resulting sampling distribution, a distribution of a statistic over repeated samples

Sampling Distribution Mean and SD:

Simple Random Sampling: A simple random sample (SRS) of size n is produced by a scheme which ensures that each subgroup of the population of size n has an equal probability of being chosen as the sample.

Stratified Random Sampling: Divide the population into "strata". There can be any number of these. Then choose a simple random sample from each stratum. Combine those into the overall sample. That is a stratified random sample. (Example: Church A has 600 women and 400 women as members. One way to get a stratified random sample of size 30 is to take a SRS of 18 women from the 600 women and another SRS of 12 men from the 400 men.)

Multi-Stage Sampling: Sometimes the population is too large and scattered for it to be practical to make a list of the entire population from which to draw a SRS. For instance, when the a polling organization samples US voters, they do not do a SRS. Since voter lists are compiled by counties, they might first do a sample of the counties and then sample within the selected counties. This illustrates two stages. In some instances, they might use even more stages. At each stage, they might do a stratified random sample on sex, race, income level, or any other useful variable on which they could get information before sampling.

How does one decide which type of sampling to use?

The formulas in almost all statistics books assume simple random sampling. Unless you are willing to learn the more complex techniques to analyze the data after it is collected, it is appropriate to use simple random sampling. To learn the appropriate formulas for the more complex sampling schemes, look for a book or course on sampling.

Stratified random sampling gives more precise information than simple random sampling for a given sample size. So, if information on all members of the population is available that divides them into strata that seem relevant, stratified sampling will usually be used.

If the population is large and enough resources are available, usually one will use multi-stage sampling. In such situations, usually stratified sampling will be done at some stages.

How do we analyze the results differently depending on the different type of sampling?

The main difference is in the computation of the estimates of the variance (or standard deviation). An excellent book for self-study is A Sampler on Sampling, by Williams, Wiley. In this, you see a rather small population and then a complete derivation and description of the sampling distribution of the sample mean for a particular small sample size. I believe that is accessible for any student who has had an upper-division mathematical statistics course and for some strong students who have had a freshman introductory statistics course. A very simple statement of the conclusion is that the variance of the estimator is smaller if it came from a stratified random sample than from simple random sample of the same size. Since small variance means more precise information from the sample, we see that this is consistent with stratified random sampling giving better estimators for a given sample size.

Hypothesis:

Hypothesis testing is a statistical method that is used in making statistical decisions using experimental data. Hypothesis Testing is basically an assumption that we make about the population parameter.

Statistical decision for hypothesis testing:

In statistical analysis, we have to make decisions about the hypothesis. These decisions include deciding if we should accept the null hypothesis or if we should reject the null hypothesis. Every test in hypothesis testing produces the significance value for that particular test. In Hypothesis testing, if the significance value of the test is greater than the predetermined significance level, then we accept the null hypothesis. If the significance value is less than the predetermined value, then we should reject the null hypothesis. For example, if we want to see the degree of relationship between two stock prices and the significance value of the correlation coefficient is greater than the predetermined significance level, then we can accept the null hypothesis and conclude that there was no relationship between the two stock prices. However, due to the chance factor, it shows a relationship between the variables.

Hypothesis Testing

We have to follow six basic steps to correctly set up and perform a hypothesis test. For example, the manager of a pipe manufacturing facility must ensure that the diameters of its pipes equal 5cm. The manager follows the basic steps for doing a hypothesis test.

NOTE

You should determine the criteria for the test and the required sample size before you collect the data.

First, the manager formulates the hypotheses. The null hypothesis is: The population mean of all the pipes is equal to 5 cm. Formally, this is written as: H0: = 5

Then, the manager chooses from the following alternative hypotheses:

Condition to test

Alternative Hypothesis

The population mean is less than the target.

one sided: < 5

The population mean is greater than the target.

one sided: > 5

The population mean differs from the target.

two sided: != 5

Because they need to ensure that the pipes are not larger or smaller than 5 cm, the manager chooses the two-sided alternative hypothesis, which states that the population mean of all the pipes is not equal to 5 cm. Formally, this is written as H1: 5

The manager uses a power and sample size calculation to determine how many pipes they need to measure to have a good chance of detecting a difference of 0.1 cm or more from the target diameter.

The manager selects a significance level 0.05, which is the most commonly used significance level.

They collect a sample of pipes and measure their diameters.

After they perform the hypothesis test, the manager obtains a p-value of 0.004. The p-value is less than the significance level of 0.05.

The manager rejects the null hypothesis and concludes that the mean pipe diameter of all pipes is not equal to 5cm.