# Assignment: The Sampling Distribution

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Discussion3: Please discuss, elaborate and give example on the topic below. Please use only the reference I attach. Please be careful with grammar and spelling. No running head please.

Author: Jackson, S.L. (2017). Statistics Plain and Simple (4th ed.): Cengage Learning

Topic:

You find out that the average 10th grade math score, for Section 6 of the local high school, is 87 for the 25 students in the class. The average test score for all 10th grade math students across the state is 85 for 1,800 students. The standard deviation for the state is 3.8.

Answer the following questions:

· What z score do you calculate?

· What is the area between the mean and the z score found in Appendix A of the textbook?

· What does this mean about the probability of this test score difference occurring by chance? Is it

less than 0.05?

Reference

The z Test: What It Is and What It Does

The Sampling Distribution

The Standard Error of the Mean

Calculations for the One-Tailed z Test

Interpreting the One-Tailed z Test

Calculations for the Two-Tailed z Test

Interpreting the Two-Tailed z Test

Statistical Power

Assumptions and Appropriate Use of the z Test

Confidence Intervals Based on the z Distribution

Review of Key Terms

Module Exercises

Critical Thinking Check Answers

Module 10: The Single-Sample t Test

The t Test: What It Is and What It Does

Student’s t Distribution

Calculations for the One-Tailed t Test

The Estimated Standard Error of the Mean

Interpreting the One-Tailed t Test

Calculations for the Two-Tailed t Test

Interpreting the Two-Tailed t Test

Assumptions and Appropriate Use of the Single-Sample t Test

Confidence Intervals Based on the t Distribution

Review of Key Terms

Module Exercises

Critical Thinking Check Answers

Chapter 5 Summary and Review

Chapter 5 Statistical Software Resources

In this chapter, we continue our discussion of inferential statistics—procedures for drawing conclusions about a population based on data collected from a sample. We will address two different statistical tests: the z test and t test. After reading this chapter, engaging in the Critical Thinking checks, and working through the problems at the end of each module and at the end of the chapter, you should understand the differences between the two tests covered in this chapter, when to use each test, how to use each to test a hypothesis, and the assumptions of each test.

Learning Objectives

•Explain what a z test is and what it does.

•Calculate a z test.

•Explain what statistical power is and how to make statistical tests more powerful.

•List the assumptions of the z test.

•Calculate confidence intervals using the z distribution.

The z Test: What It Is and What It Does

The  z test  is a parametric statistical test that allows you to test the null hypothesis for a single sample when the population variance is known. This procedure allows us to compare a sample to a population in order to assess whether the sample differs significantly from the population. If the sample was drawn randomly from a certain population (children in academic after-school programs) and we observe a difference between the sample and a broader population (all children), we can then conclude that the population represented by the sample differs significantly from the comparison population.

z test A parametric inferential statistical test of the null hypothesis for a single sample where the population variance is known.

Let’s return to our example from the previous module and assume that we have actually collected IQ scores from 75 students enrolled in academic after-school programs. We want to determine whether the sample of children in academic after-school programs represents a population with a mean IQ greater than the mean IQ of the general population of children. As stated previously, we already know μ (100) and σ (15) for the general population of children. The null and alternative hypotheses for a one-tailed test are:

In Module 6 we learned how to calculate a z score for a single data point (or a single individual’s score). To review, the formula for a z score is:

z=X−μσz=X−μσ

where

X = each individual score

μ = the population mean

σ = the population standard deviation

Remember that a z score tells us how many standard deviations above or below the mean of the distribution an individual score falls. When using the z test, however, we are not comparing an individual score to the population mean. Instead, we are comparing a sample mean to the population mean. We therefore cannot compare the sample mean to a population distribution of individual scores. We must compare it instead to a distribution of sample means, known as the sampling distribution.

The Sampling Distribution

If you are becoming confused, think about it this way. A  sampling distribution  is a distribution of sample means based on random samples of a fixed size from a population. Imagine that we have drawn many different samples of some size (say 75) from the population (children whose IQ can be measured). For each sample that we draw, we calculate the mean; then we plot the means of all the samples. What do you think the distribution will look like? Well, most of the sample means will probably be similar to the population mean of 100. Some of the sample means will be slightly lower than 100; some will be slightly higher than 100; and others will be right at 100. A few of the sample means, however, will not be similar to the population mean. Why? Based on chance, some samples will contain some of the rare individuals with either very high IQ scores or very low IQ scores. Thus, the means for those samples will be much lower than 100 or much higher than 100. Such samples, however, will be few in number. Hence, the sampling distribution (the distribution of sample means) will be normal (bell-shaped), with most of the sample means clustered around 100 and a few sample means in the tails or the extremes. Therefore, the mean for the sampling distribution will be the same as the mean for the distribution of individual scores (100).

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