AP Statistics Chap 10-1 Confidence Intervals

AP Statistics Chap 10-2 Confidence Intervals Population Mean σ Unknown (Lock 6.5) Confidence Intervals Population Proportion (Lock 6.1) σ Known (Lock 5.2)

Central Limit Theorem AP Statistics Chap 10-3 Normal if: Asymmetric: 30+ Symmetric: 15+ Normal: >1

AP Statistics Chap 10-4 Confidence Interval for μ ( σ Known) Assumptions Population standard deviation σ is known Population is normally distributed If population is not normal, use large sample Confidence interval estimate

AP Statistics Chap 10-5 Common Levels of Confidence Commonly used confidence levels are 90%, 95%, and 99% Confidence Level Confidence Coefficient, z value, % 90% 95% 98% 99% 99.8% 99.9%

AP Statistics Chap 10-6 Finding the Critical Value Consider a 95% confidence interval: z.025 = -1.96z.025 = 1.96 Point Estimate Lower Confidence Limit Upper Confidence Limit z units: x units: Point Estimate 0

Confidence Intervals 90% Samples 95% Samples 99% Samples    x    x  x_ XXXX    x    x    x    x   X  =  ± Z   x

AP Statistics Chap 10-8 Interval and Level of Confidence Confidence Intervals Intervals extend from to 100(1-  )% of intervals constructed contain μ ; 100  % do not. Sampling Distribution of the Mean x x1x1 x2x2

AP Statistics Chap 10-9 Margin of Error(MoE) Margin of Error (e): the amount added and subtracted to the point estimate to form the confidence interval Example: Margin of error for estimating μ, σ known:

AP Statistics Chap Factors Affecting MoE Data variation, σ : e as σ Sample size, n : e as n Level of confidence, 1 -  : e if 1 - 

AP Statistics Chap Example 1 A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is.35 ohms. Determine a 95% confidence interval for the true mean resistance of the population.

AP Statistics Chap Example 1 A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is.35 ohms. (continued) Solution: 2.20±1.96(.35/sqrt(11) 2.20±.2068 [1.99, ]

Z-Interval w/ Calculator AP Statistics Chap 10-13

AP Statistics Chap Interpretation We are 95% confident that the true mean resistance is between and ohms Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean An incorrect interpretation is that there is 95% probability that this interval contains the true population mean. (This interval either does or does not contain the true mean, there is no probability for a single interval)

Thinking Challenge You’re a Q/C inspector for Gallo. The  for 2- liter bottles is.05 liters. A random sample of 100 bottles shows  X = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles? 2 liter © T/Maker Co. 2 liter

Confidence Interval Solution

AP Statistics Chap Computer Simulation

AP Statistics Chap Example 2 Tim Kelley weighs himself once a week for several years. Last month he weighed himself 4 times with an average of Examination of Tim’s past data reveals that over relatively short periods of time, his weight measurements are approximately normal with a standard deviation of about 3. Find a 90% confidence interval for his mean weight for last month. Then, find a 99% confidence interval.

AP Statistics Chap Example 2 Suppose Tim had only weighed himself once last month and that his one observation was x=190.5 (the same as the mean before). Estimate µ with 90% confidence. (continued)

AP Statistics Chap Example 2 Tim wants to have a margin of error of only 2 pounds with 95% confidence. How many times must he weigh himself to achieve this goal? (continued)

AP Statistics Chap Confidence Intervals with Unknown Population Standard Deviation

AP Statistics Chap Confidence Intervals Population Mean σ Unknown Confidence Intervals Population Proportion σ Known

AP Statistics Chap If the population standard deviation σ is unknown, we can substitute the sample standard deviation, s This introduces extra uncertainty, since s is variable from sample to sample So we use the t distribution instead of the normal distribution Confidence Interval for μ ( σ Unknown)

AP Statistics Chap Assumptions Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample Use Student’s t Distribution Confidence Interval Estimate Confidence Interval for μ ( σ Unknown) (continued)

AP Statistics Chap Student’s t Distribution The t is a family of distributions The t value depends on degrees of freedom (d.f.) Number of observations that are free to vary after sample mean has been calculated d.f. = n - 1

AP Statistics Chap If the mean of these three values is 8.0, then x 3 must be 9 (i.e., x 3 is not free to vary) Degrees of Freedom (df) Idea: Number of observations that are free to vary after sample mean has been calculated Example: Suppose the mean of 3 numbers is 8.0 Let x 1 = 7 Let x 2 = 8 What is x 3 ? Here, n = 3, so degrees of freedom = n -1 = 3 – 1 = 2 (2 values can be any numbers, but the third is not free to vary for a given mean)

AP Statistics Chap Student’s t Distribution t 0 t (df = 5) t (df = 13) t-distributions are bell- shaped and symmetric, but have ‘fatter’ tails than the normal Standard Normal (t with df =  ) Note: t z as n increases

AP Statistics Chap Student’s t Table Upper Tail Area df t The body of the table contains t values, not probabilities Let: n = 3 df = n - 1 = 2  =.10  /2 =.05  /2 =.05

AP Statistics Chap t distribution values With comparison to the z value Confidence t t t z Level (10 d.f.) (20 d.f.) (30 d.f.) ____ Note: t z as n increases

AP Statistics Chap Example A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ CI = , The confidence interval is:

Thinking Challenge You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time?

Confidence Interval Solution*  X = 3.7 S = n = 6, df = n - 1 = = 5 S /  n = 3.99 /  6 = t.05,5 = (2.015)(1.592)  (2.015)(1.592)  6.908

AP Statistics Chap Approximation for Large Samples Since t approaches z as the sample size increases, an approximation is sometimes used when n  30: Correct formula Approximation for large n

AP Statistics Chap Determining Sample Size The required sample size can be found to reach a desired margin of error (e) and level of confidence (1 -  ) Required sample size, σ known: 2 /2 2 ME σz σz n         2 2

AP Statistics Chap Required Sample Size Example If  = 45, what sample size is needed to be 90% confident of being correct within ± 5? (Always round up) So the required sample size is n = 220

AP Statistics Chap If σ is unknown If unknown, σ can be estimated when using the required sample size formula Use a value for σ that is expected to be at least as large as the true σ Select a pilot sample and estimate σ with the sample standard deviation, s

AP Statistics Chap Chapter 10 Confidence Intervals for Population Proportions

AP Statistics Chap Confidence Intervals Population Mean σ Unknown Confidence Intervals Population Proportion σ Known

AP Statistics Chap Confidence Intervals for the Population Proportion, p An interval estimate for the population proportion ( p ) can be calculated by adding an allowance for uncertainty to the sample proportion ( )

AP Statistics Chap Confidence Intervals for the Population Proportion, p Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation We will estimate this with sample data: (continued)

AP Statistics Chap Confidence interval endpoints Upper and lower confidence limits for the population proportion are calculated with the formula where z is the standard normal value for the level of confidence desired p is the sample proportion n is the sample size

AP Statistics Chap Example A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers

AP Statistics Chap Example A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers (continued)

AP Statistics Chap Interpretation We are 95% confident that the true percentage of left-handers in the population is between 16.51% and 33.49%. Although this range may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.

AP Statistics Chap Changing the sample size Increases in the sample size reduce the width of the confidence interval. Example: If the sample size in the above example is doubled to 200, and if 50 are left-handed in the sample, then the interval is still centered at.25, but the width shrinks to.19 …….31

Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p.

Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p.

Thinking Challenge You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?

Confidence Interval Solution*

AP Statistics Chap Finding the Required Sample Size for proportion problems Solve for n: Define the margin of error: p can be estimated with a pilot sample, if necessary (or conservatively use p =.50)

AP Statistics Chap What sample size...? How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence? (Assume a pilot sample yields p =.12)

AP Statistics Chap What sample size...? Solution: For 95% confidence, use Z = 1.96 E =.03 p =.12, so use this to estimate p So use n = 451 (continued)

AP Statistics Chap Chapter Summary Illustrated estimation process Discussed point estimates Introduced interval estimates Discussed confidence interval estimation for the mean (σ known) Addressed determining sample size Discussed confidence interval estimation for the mean (σ unknown) Discussed confidence interval estimation for the proportion

AP Statistics Chap Confidence Intervals End of Basic Confidence Intervals