Intro to Confidence Intervals SECTION 10.1 1 Confidence Intervals - PDF document

Confidence Intervals Slides.notebook December 22, 2015 Intro to Confidence Intervals SECTION 10.1 1

Confidence Intervals Slides.notebook December 22, 2015 Definitions • Statistical Inference: • estimate of the characteristics of a population • derived from the analysis of a sample drawn from the population • provides methods of drawing conclusions about a population from sample data • population: everyone • parameter: a number from a population • Symbols­­­ • sample: a part of the population • statistic: a number from a sample • Symbols­­­ 2

Confidence Intervals Slides.notebook December 22, 2015 NOTE TO SELF The Central Limit theorem plays an important role in statistical inference. 2 types of statistical inference: Confidence intervals (10.1) Tests of significance (10.2) 3

Confidence Intervals Slides.notebook December 22, 2015 • You want to estimate the mean SAT Math score for the more than 350,000 high school seniors in California. • You give the test to a simple random sample (SRS) of 500 CA seniors. • RECALL: • The central limit theorem tells us that the mean, x, of 500 scores has a distribution that is close to normal. 4

Confidence Intervals Slides.notebook December 22, 2015 • The mean of this normal sampling distribution is the same as the unknown mean, , of the entire population. • The standard deviation of x for an SRS of 500 students is , where is the standard deviation of individual SAT MATH scores among all CA high school seniors. 5

Confidence Intervals Slides.notebook December 22, 2015 Statistical Confidence • The 68­95­99.7 rule says that in 95% of all samples, the mean score x for the sample will be within 2 standard deviations (9 points in this example) of the population mean score • Therefore, in 95% of all samples, the unknown lies between x – 9 and x + 9. 6

Confidence Intervals Slides.notebook December 22, 2015 Confidence Interval has the form: estimate + margin of error Estimate: guess for unknown parameter (usually μ) We will use X here MOE: how accurate we believe guess is based on variability of estimate 7

Confidence Intervals Slides.notebook December 22, 2015 Confidence Level, C Gives probability that the interval will capture true population mean, μ, in repeated samples “C” will be expressed in decimal form Note: we generally want a confidence level of .9 or 90% or higher 8

Confidence Intervals Slides.notebook December 22, 2015 • This shows the result of drawing 25 SRSs from the same population and calculating a 95% confidence interval from each sample. • The center x of each interval is marked by a dot. • The arrows on either side of the dot span the confidence interval. • CONCLUSION: 95% of all samples give an interval that contains the population mean . 9

Confidence Intervals Slides.notebook December 22, 2015 Formula for Confidence Interval Choose an SRS of size n from a population having unknown mean μ and known standard deviation σ. A level C confidence interval for μ is estimate + margin of error • Everything will be given in the problem except z* • z* depends on the confidence level you choose • z* is the value with area C between –z* and z* under the standard normal curve. 10

Confidence Intervals Slides.notebook December 22, 2015 Most common confidence levels: C Tail Area z* 90% 0.05 1.645 95% 0.025 1.960 99% 0.005 2.576 Calculator can find z* values: 11

Confidence Intervals Slides.notebook December 22, 2015 What does 95% confidence mean? • We are 95% confident that the true population mean is captured in the interval OR • We are using a procedure that captures the true population mean 95% of the time. 12

Confidence Intervals Slides.notebook December 22, 2015 Example 1 I sample 30 cats. X = 6 lbs. I know that σ = 3 lbs. • a) Find a 90% CI. • b) Find a 95% CI. • c) Find a 99% CI. **Express what these confidence intervals indicate in terms of the problem.** e) Explain why the intervals get wider as your confidence level increases. 13

Confidence Intervals Slides.notebook December 22, 2015 CALCULATOR!!!! • STAT • TEST • Z­INTERVAL (#7) • STATS • Plug info in and calculate. • Show 14

Confidence Intervals Slides.notebook December 22, 2015 Example 2 Heights are normally distributed. I measure the heights of 25 randomly selected students. The average is 66 inches. I know that the population S.D. (σ) is 4 inches. • a) Find a 90% CI. • b) Find a 95% CI. • c) Find a 97% CI. • d) Find a 99.9% CI 15

Confidence Intervals Slides.notebook December 22, 2015 Example 3 SAT verbal scores are normally distributed with σ = 50. 25 students are selected randomly and their average score is 492. • a) Find a 99% Confidence Interval. • If 100 students are selected and their average is still 492, find the new 99% Confidence Interval. 16

Confidence Intervals Slides.notebook December 22, 2015 Example 4 The following 10 bunny weights are collected. 7 8 9 8.5 10 11 8.5 9.5 10.5 10 Find a 90% confidence interval for bunny weights based on this sample! Assume σ = 1.5. 17

Confidence Intervals Slides.notebook December 22, 2015 In Summary Increasing my confidence level will _________________ my margin of error, and therefore also ____________ my interval width. Increasing my sample size will _______________ my margin of error. If we want to cut our M.O.E. in half, what should we do to our sample size? 18

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