University:
Montgomery College
Course:
MATH 117: Elements of Statistics
Academic year:

2020
Views:

229

Pages:

5
Author:

Corey Garrison

Unit 5 - Binomial Distributions Chapter 18 - Sampling Distributions Day 4: Sampling distributions, sample means, Central Limit Theorem Today you will be able to calculate probabilities using a normal model with sampling distributions. Warm ups: Take out your warm up packets... MC (part 1) #16, 17, 18, 24, 25 Homework Question In a large survey of US adults, the professor has each person toss a coin 10 times and calculate the proportion of his or her tosses that were heads. The students then import their results, and the professor plots a histogram of these sample proportions. (a) What shape would you expect this histogram to be? Why? (b) Where do you expect the histogram to be centered? (c) How much variability would you expect among these proportions? (d) Explain why a Normal model should not be used here. Sampling Distributions Bias vs. Variability: Bias is the accuracy of a statistic. Variability is the precision of a statistic. Sampling Distribution of Sample Means You pick 10 SRSs of size n. SRSs of size n = ... The height of females is normally distributed with a mean of 64.5 inches and a standard deviation of 2.5 inches. (a) What is the probability that one randomly selected woman is more than 66 inches? P(X > 66) = .274 N(64.5, 2.5) normalcdf (66, 1000, 64.5, 2.5) (b) The height of females is normally distributed with a mean of 64.5 inches and a standard deviation of 2.5 inches. What is the probability that the average height of 10 randomly selected women is more than 66 inches? μ_x̄ = 64.5 σ_x̄ = 2.5 / √10 = .791 N(64.5, .791) P(x̄ > 66) = .029 (c) What is the probability that the average height of 100 randomly selected women is more than 66 inches? μ_x̄ = 64.5 σ_x̄ = 2.5 / √100 = .25 N(64.5, .25) P(x̄ > 66) = .0000000000099 2. The scores of students on the ACT college entrance examination in a recent year had the normal distribution with μ = 18.6 and σ = 5.2. (a) What is the probability that a single student chosen from all those taking the test scores 21 or higher? (b) Now take an SRS of 50 students who took the test. What are the mean and standard deviation of 1 the sampling distribution of their mean score? (c) What is the probability that the mean score of these students is 21 or higher? What if the distribution of the population is NOT normal to begin with? Let's find out... What if the population is skewed right? Same thing as before! What if the population is uniformly distributed? Same thing as before! What if the population is some "crazy" distribution? Same thing as before! Central Limit Theorem (CLT) MOST IMPORTANT THEOREM Even if the distribution of the population is NOT normal, as n increases, the distribution of the sample mean gets closer to a normal distribution! When n is at least 25, the sampling distribution is approximately normally distributed with... μ_x̄ = μ σ_x̄ = σ / √n