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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 descriptives and sampling distributions. Descriptives 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.

Response 1

Random Sampling is considered the easiest way of sampling ways wherein all participants of a given populace have the same possibility of being selected for the example group. One strategy of assuring a random example is to allocate amounts of the populace and select the sample through unsystematic assortment of numbers

Advantages:

  • Offers Sample of the Whole Population of a Region: This is one great advantage of this kind of survey method. This might be ideal to utilize to understand what if one who likes to understand what people all through the place are planning, like getting ideas on a problem to assist politician take a position on a law for latest improvements.
  • You Can Get Feedback: Another benefit the random sampling could provide is that you can get feedback from a person which is really utilizing a service. You can also control for randomness through having a skilled interviewer choose the one passing the interviewer or the person the interviewers passes upon concluding an interview with the past selected individual.

Disadvantages:

  • Biased Results: This is one of the major disadvantages of random sampling. Individual prejudices might also creep into the information, as an assessor might not spread the questionnaires to specific group of people. These aspects often lead to twisted information gathering, rendering the information not valuable for monitoring trends all through the whole population.
  • Laborious and Time Consuming: This is also one major drawback of the random sampling as one try to get a sampling of all and sundry in the entire population. Also while many organizations doing international surveys search to utilize multi stage as well as it make the process of colleting samples more manageable, there could be issues doing this.

To get the ideal random sample, the random should be finite and the whole members of the populace should be determined and listed so as to prevent bias.

Reference:

Occupytherapy. (2014). Advantages and disadvantages of random sampling. Retrieved from https://occupytheory.org/advantages-and-disadvanta…

Response 2

“Probability sampling, also known as, random sampling, requires that every member of the study population has an equal and independent opportunity to be chosen for inclusion in a study” (Grove, 2013). Random sampling is achieved through randomly selecting members from a group, and this is commonly accomplished via computer programing (Grove,2013).Why is random sampling important? Researchers gather a small sample from a larger populace to paint an accurate picture of the larger group.Random sampling is easy to use and does not require the alienation of subjects. Examples of random sampling techniques are as simple as drawing straws, picking a number, and drawing from a brown paper bag. Random sampling is a way to remove bias in sample selection, but random sampling has its limitations (Annenberg,2017).

Laerd (2012) identifies numerous limitations linked to random samples. These limitations include cost, time limitations, and individual prejudices (Laerd, 2012). Identifying the populace of a random sample is difficult, and sample lists may not be readily available, or they could be protected by privacy policies (Laerd,2012). This means researchers may not have the means to identify all members of the sample. Furthermore, biased results can significantly alter a random sample.

Annenberg Learner. 2017. Retrieved from: https://www.learner.org/courses/learningmath

Laerd Dissertation. 2012. Simple Random Sampling. Retrieved from:

http://dissertation.laerd.com/simple-random-sampli…

Grove, Susan K., Daisha Cipher. Statistics for Nursing Research: A Workbook for Evidence-

Based Practice, 2nd Edition . Saunders, 022016. VitalBook file.

Response 3

Radom sampling is important in that the study remains unbiased. Random sampling is not necessarily a true depiction of the general population but excludes a bias interference. Problems that may occur in random sampling might be in a situation of requesting feedback from a random sample of people. Information may be skewed in that those that feel strongly about the question being provided. For example, say a questionnaire was sent out to the general public asking if they mistrust car salesman. Those that have strong feelings of mistrust may respond more than someone that has no mistrust. A way this can be avoid is a good rating system. Asking some rate their feelings af car salesmen can bring in a larger number and more random respondants than asking a single question. We as a society tend to report the bad news but forgo the good. This is what is called sampling error.

The Economic Times (2017) Definition of Random Sampling. Retrieved from https://economictimes.indiatimes.com/definition/ra…

 
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