# The distribution of sample means over repeated

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Chapter 5 / Exercise 45
Elementary and Intermediate Algebra
Tussy/Gustafson
Expert Verified
- the distribution of sample means over repeated sampling from one population Goal: -estimate a population mean given sample mean(s) -as increase sample size (or the number of samples), approximate a normal curve 2. Central Limit Theorem **Central Limit Theorem - Given a population with mean and variance 2 , the sampling distribution of the mean will have a mean equal to and a variance equal to 2 / N. The distribution will approach the normal distribution as N, the sample size, increases. -the rate at which the sampling distribution of the mean approaches normal is a function of the shape of the parent population: a) -if the population is normal, the sampling distribution of the mean will be normal regardless of N b) -if the population is symmetric but non-normal, the sampling distribution of the mean will be nearly normal even for quite small sample sizes c) -if the population is markedly skewed, sample sizes of 30 or more are required before the means approximate a normal distribution Examples: Sampling Distributions 1. Use your z-tables to find the critical z-values: a. = .05 2-tails b. = .05 1-tail c. = .01 2-tails d. = .01 1-tail 2. Use your t-tables to find the critical t-values: a. n = 25 = .05 2-tails b. n = 25 = .01 1-tail c. n = 30 = .01 2-tails d. n = 30 = .05 1-tail
3. The national average for the verbal section of the Graduate Record Exam (GRE) is 500 with a standard deviation of 100. A researcher uses a sampling distribution made up of samples of 100. a. According to the Central Limit Theorem, what is the mean of the sampling distribution of means? b. According to the Central Limit Theorem, what is the standard error of the mean?
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Chapter 5 / Exercise 45
Elementary and Intermediate Algebra
Tussy/Gustafson
Expert Verified
PSY1110: Modules 9 & 10 (Combined) Course Notes Page 5 3b. 10 Hypothesis Testing (steps 4 to 8): 1) Z-Score Hypothesis Testing with Single X-Value z-score formula X ***This formula is used when both and are known 2) Z-Score Hypothesis Testing with Sample of X-Values z-test formula z X   x , where x Examples (for the z-test): A. Overall, Honda Accords get 28 mpg (miles per gallon) (μ = 28) on the highway with a standard deviation of 1.3 miles (σ = 1.3). Someone believes a gas additive will improve the mileage? After adding the additive, a car gets 31 mpg. Does the additive improve mileage? Set alpha equal to 0.05. z n