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BA 253: Business Statistics Today Sampling Distributions Confidence Intervals 9/30/15 Fri More CI ICE 5 ↑ Descriptive Statistics: Summarize Data ↑ ↓ Inferential Statistics: Estimate Population from Sample ↓ Chapter 7: Sampling and Sampling Distributions Ex: Collect n data, calculate sample mean = x = point estimate. Use simple random sample. Is x ≈ μ??? That is the question! The approximation gets better as the sample size, n, increases. How is x distributed? In other words, what is the probability distribution of the sample mean? Guess = _________________ Central Limit Theorem For large sample sizes (typically, n ≥ 30), the sample mean x is approximately normally distributed. CLT is true regardless of the original distribution of the data. As n↑, the standard deviation decreases. (That is, x gets closer to μ.) This is why the normal distribution is so common and so important. Sampling Distribution of x The sample mean, x , has mean μ, standard deviation and is n o Normally distributed if the underlying distribution is Normal. o Approximately Normally Distributed whatever the underlying distribution is if n ≥ 30. o As the sample size increases, n ↑, the sample mean gets closer and closer to the population mean, x ≈ μ. Show Galton board video and Plinko video… Ex: Normal with mean 100 and sd 20. How close is your result to 100? Is one data point enough? No. Get 5 numbers, then average. Closer. On Excel, get 30, 100, 1000 numbers and then average. As sample size increases, the approximation gets better. Confidence Intervals for Mean Ex: How much do FLC students spend on gas per week? Survey 50 students and get: x = $37, s = $19. Is μ ≈ x = $37? How close is the sample mean to the population mean? With 95% certainty, what is the population mean? o Show/discuss formula. o (Despite having only surveyed 50 people) we are 95% certain that the population mean µ (the average for the entire population) is somewhere in the range from [$32, $42]. Want to reduce the size of the interval? Collect more data! Common Confidence Levels: Confidence 90% 95% 98% 99% z-score ± 1.645 ± 1.96 ± 2.33 ± 2.575 Show where these come from. Sample Size Determination: What if we want the error to only be ± $2.5? How many samples n do we need? o E = ………. n = ………. = 222 o What if we want ± $1??? n = 1387. Error ± $5 ± $2.5 ± $1 ± $0.1 Sample Size n = 50 n = 222 n = 1387 n = 138682 Ex: How many texts do college students send daily? Collect data. n < 30, but assume Normal anyway, probably reasonable. Determine a 95% confidence interval. What sample size would be needed to know within ± 10??? Confidence Interval for Proportions: Ex: Same survey as before. With higher gas prices, have you considered switching to a more fuel efficient vehicle? p = 37/50 = 74% = 0.74 Is p = p = 74%? With 95% certainty, what is p? Point Estimate: p = 74%. Confidence Interval: p ± E = …… = 74% ± 12% = (61%, 85%) How many data necessary to know p within ± 5%? E = ……, solve for n, = 303 What if we didn’t know p ? Then assume p = ½. Sample Size Determination: At 95% confidence, how many samples n necessary to know a proportion within ± 5%? ± 1%? Quick estimate Fill in……….. Point Estimate Confidence Interval How many samples? Mean Proportion