Read the lecture and respond to the discussion questions with reference

Populations and Sampling Distributions

Introduction

In the previous topic the different techniques for obtaining a sample were discussed, as well as distributions and some descriptive statistics. This lecture will continue to discuss descriptive and sampling distributions. Descriptive and the distribution will be used to calculate probabilities. Probability is the likelihood that an event will occur.

Normal Distribution

The normal distribution is the distribution that is most commonly worked with in scientific research. The curve is bell shaped with its highest point over the mean. The curve is symmetrical about the mean. The curve will approach the horizontal axis but never quite touch it. The normal distribution is completely described by the mean, and standard deviation of the data set. The most common normal distribution is the standard normal. The standard normal is represented by the letter z, has a mean of 0, and a standard deviation of 1.

Z-scores

The z-score is another descriptive. A z-score tells the location of an individual score as it relates to the mean of the data. To find the z-score of an individual score, subtract the mean, and then divide by the standard deviations.

The z-score tells how many standard deviations the score is away from the mean. The z-score equation will give the position of any score in the distribution.

Normal Probabilities

Finding the probabilities for the normal distribution is equivalent to finding the area under the normal curve. To find this area, the standard normal table is used. Since not all normal distributions are standard normal, it is necessary to convert them to the standard normal in order to use the table. To perform this conversion, use the z-score equation. Once the scores are standardized, the probabilities can be found using the standard normal table. An example of this can be found in the Visual Learner: Statistics.

Sampling Distributions

Distributions of populations of scores have been discussed. However, a single score does not accurately represent the population. It is time to look at the distribution of the sample means. The distribution of the sample means is described by the central limit theorem, which states that for a random variable X, with a mean of and a standard deviation of , the sample mean will have a mean equal to the population mean and a standard deviation equal to the standard error. The standard error is the population standard deviation, divided by the square root of the sample size. The distribution of the sample mean will follow a normal distribution if the data is normally distributed, or if the sample size is greater than 30 (Brase & Brase, 2010). Once the distribution of the sample means is identified, then probabilities can be calculated based on the sample means. Examples of this are shown in the Visual Learner: Statistics.

Conclusion

An important distribution was focused on in this topic. The normal distribution is a major portion of research. Many of the tests that will be discussed in the future require the use of the normal distribution. The techniques discussed here will be used to perform the hypothesis tests that will be discussed in the next topic.

References

Brase, C., & Brase, C. (2010). Understanding basic statistics (5th ed.). Belmont, CA: Cengage Learning.

Discussion 1

Explain the importance of random sampling. What problems/limitations could prevent a truly random sampling and how can they be prevented?

Discussion 2

Explain each sampling technique discussed in the "Visual Learner: Statistics" in your own words, and give examples of when each technique would be appropriate.