COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick

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# COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick - PowerPoint PPT Presentation

COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick. Chapter 7: Distribution of Sample Means. Key Terms: Don’t Forget Notecards. Sampling Error (p. 201) Distribution of Sample Means (p. 201)

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## COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick

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COURSE: JUST 3900

TIPS FOR APLIA

Developed By:

John Lohman

Michael Mattocks

Aubrey Urwick

Chapter 7:

Distribution of Sample Means

Key Terms: Don’t Forget Notecards
• Sampling Error (p. 201)
• Distribution of Sample Means (p. 201)
• Sampling Distribution (p. 202)
• Central Limit Theorem (p. 205)
• Expected Value of M (p. 206)
• Standard Error of M (p. 207)
• Law of Large Numbers (p. 207)
Formulas
• Standard Error of M:
• z-Score Formula:
Central Limit Theorem
• Question 1: A population has a mean of µ = 50 and a standard deviation of σ = 12.
• For samples of size n = 4, what is the mean (expected value) and the standard deviation (standard error) for the distribution of sample means?
• If the population distribution is not normal, describe the shape of the distribution of sample means based on n = 4.
• For samples of size n = 36, what is the mean (expected value) and the standard deviation (standard error) for the distribution of sample means?
• If the population distribution is not normal, describe the shape of the distribution of sample means based on n= 36.
Central Limit Theorem
• Expected Value of M: µ = 50

Standard Error of M:

• The distribution of sample means does not satisfy either criterion to be normal. It would not be a normal distribution.
• Expected Value of M: µ = 50

Standard Error of M:

• Because the sample size is greater than n = 30, the distribution of sample means is a normal distribution.
Understanding the Sampling Distribution of M
• Question 2: As sample size increases, the value of expected value of M also increases. (True or False?)
Understanding the Sampling Distribution of M
• False. The expected value of M does not depend on sample size; it will always be equal to the population mean: µ.
Understanding the Sampling Distribution of M
• Question 3: As sample size increases, the value of the standard error also increases. (True or False?)
Understanding the Sampling Distribution of M
• False. The standard error decreases as sample size increases.
• In Question 1a, the standard error was
• However in Question 1c, in which the sample size was increased from
• n = 4 to n = 36, the standard error decreased:
Using z-Scores with the Distribution of Sample Means
• Question 4: For a population with a mean of µ = 40 and a standard deviation of σ = 8, find the z-score corresponding to a sample mean of M = 44 for each of the following sample sizes.
• n = 4
• n = 16
Using z-Scores with the Distribution of Sample Means
• The standard error is , and
• The standard error is , and
Using z-Scores with the Distribution of Sample Means
• Question 5: What is the probability of obtaining a sample mean greater than M = 60 for a random sample of n = 16 scores selected from a normal population with a mean of µ = 65 and a standard deviation of σ = 20?
Using z-Scores with the Distribution of Sample Means
• The standard error is ,
• Mcorresponds to ,
• p(M > 60) = p(z > -1.00) = 0.8413 (or 84.13%)

Z = -1.00

Using z-Scores with the Distribution of Sample Means
• Question 6: A positively skewed distribution has µ = 60 and σ = 8.
• What is the probability of obtaining a sample mean greater than M = 62 for a sample of n = 4 scores?
• What is the probability of obtaining a sample mean greater than M = 62 for a sample of n = 64 scores?
Using z-Scores with the Distribution of Sample Means
• The distribution does not satisfy either of the criteria for being normal. Therefore, you cannot use the unit normal table, and it is impossible to find the probability.
• With n = 64, the distribution of sample means is nearly normal. The standard error is ,

M corresponds to

p(M > 62) = p(z > 2.00) = 0.0228 (or 2.28%)

Remember: A distribution of sample means is normal if at least one of the following

condition are met:

The population from which the samples are selected is normal,

The number of scores (n) in each sample is relatively large, around 30 or more.

Three Different Distributions
• Question 7: A population has a mean of µ = 100 and a standard deviation of σ = 15. A sample of n = 25 scores is taken with a mean of M = 101.2 and a standard deviation of s = 11.5.
• On average, how much difference should there be between the population mean and a single score selected from this population?
• On average, how much difference should there be between the sample mean and a single score selected from that sample?
• On average, how much difference should there be between the population mean and the sample mean of any sample consisting of n = 25 scores?
Three Different Distributions
• Question 7:
• σ = 15
• s = 11.5
• What effect does sample size have on the standard error?
• As sample size increases, standard error decreases. This is because large samples are more representative of the population. Thus we can expect less difference, or error.
• As sample size decreases, standard error increases. This is because smaller samples are less representative of the population. Thus we can expect a greater difference, or error.