Estimating Single Population Parameters PDF

Estimating Single Population Parameters  Chapter Goals After completing this chapter, you should be able to: Distinguish between a point estimate and a confidence interval estimate Construct and interpret a confidence interval estimate for a single populat...

Estimating Single Population Parameters  Chapter Goals After completing this chapter, you should be able to: Distinguish between a point estimate and a confidence interval estimate Construct and interpret a confidence interval estimate for a single population mean using both the z and t distributions Determine the required sample size to estimate a single population mean within a specified margin of error Form and interpret a confidence interval estimate for a single population proportion  Point and Interval Estimates A point estimate is a single number, a confidence interval provides additional information about variability Lower Upper Confidence Confidence Point Estimate Limit Limit Width of confidence interval  Point Estimates We can estimate a with a Sample Population Parameter … Statistic (a Point Estimate) Mean μ x Proportion p P̂ OrP  Confidence Intervals How much uncertainty is associated with a point estimate of a population parameter? We cannot eliminate sampling error We need a interval to quantify the uncertainty associated with our point estimate So we consider the sampling error in our decision An interval estimate provides more information about a population characteristic than does a point estimate Such interval estimates are called confidence intervals Confidence Interval Estimate An interval gives a range of values: Takes into consideration variation in sample statistics from sample to sample Based on observation from 1 sample Gives information about closeness to unknown population parameters Stated in terms of level of confidence Never 100% sure  Estimation Process Random Sample I am 95% confident that μ is between Population Mean 40 & 60. (mean, μ, is x = 50 unknown) Sample  General Formula The general formula for all confidence intervals is: Point Estimate  (Critical Value) x (Standard Error)  Confidence Level Confidence Level Confidence in which the interval will contain the unknown population parameter A percentage (less than 100%)  Confidence Level, (1-) (continue d) Suppose confidence level = 95% Also written (1 - ) =.95 A relative frequency interpretation: In the long run, 95% of all confidence intervals that can be constructed will contain the unknown true parameter A specific interval either will contain or will not contain the true parameter No probability involved in a specific interval  Confidence Intervals Confidence Intervals Population Population Mean Proportion σ Known σ Unknown  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 𝝈 𝒙 ± 𝒛 𝜶/ 𝟐 √𝒏 Point Estimate  (Critical Value) x (Standard Error)  Finding the Critical Value given the Confidence Level Consider a 95% confidence interval: z α/2 1.96 1   .95 α α .025 .025 2 2 z units: z.025= -1.96 0 z.025= 1.96 Lower Upper x units: Confidence Point Estimate Confidence Limit Limit  Interval and Level of Confidence Sampling Distribution of the Mean /2 1  /2 x Intervals μx μ extend x1 fromσ x2 100(1-)% x  z /2 of intervals n constructed to σ contain μ; x  z /2 n 100% do Confidence Intervals not. See simulation: http://www.rossmanchance.com/applets/ConfSim.html  Margin of Error 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: σ σ x z /2 e z /2 n n  Factors Affecting Margin of Error σ e z /2 n Data variation, σ : e as σ Sample size, n : e as n Level of confidence, 1 -  : e if 1 -   Example: Confidence Interval Estimate for when is known Assuming that the population distribution for credit card balances is normal with a population standard deviation equal to $400, construct and interpret a 95% confidence interval estimate for the population mean balance if a random sample of n = 100 accounts have a sample mean equal to $2,000. Point Estimate (Critical Value)(Standard Error) 𝜎 𝑥 ± 𝑧 𝛼/ 2 known) √𝑛  Example: Confidence Interval Estimate for when is known Confidence Interval: 𝜎 Confidence Interval 𝑥 ± 𝑧 𝛼/ 2 √𝑛 (normal table) $ 400 $ 2,000 ± 1.96 √ 100Interpretation: Based on the sample data, $2,000 ±$78.40 we are 95% confident that the population (or true) ----- mean is between $1,921.60 and $2,078.40.  Example: Confidence Interval Estimate for when is known Confidence Interval: What is the margin of 𝜎 error? 𝑥 ± 𝑧 𝛼/ 2 √𝑛 $ 400 $ 2,000 ± 1.96 √ 100 Interpretation: The margin of error $2,000 ±$78.40 indicates that the sample mean of $2,000 ----- is within $78.40 of the true population mean.  Example: Confidence Interval Estimate for when is known 𝜎 𝜎 𝐶𝐼 : 𝑥 ± 𝑧 𝑒= 𝑧 Impact of sample size n: √𝑛 √𝑛 n=100: n=200: n=400: Impact of confidence level: 90% CI: 95% CI: 99% CI:  Confidence Intervals Confidence Intervals Population Population Mean Proportion σ Known σ Unknown  Confidence Interval for μ (σ Unknown) If the population standard deviation σ is unknown, we can substitute the sample standard deviation, s: This introduces extra uncertainty, since s will vary from sample to sample So we use the Student’s t distribution instead of the normal distribution  Confidence Interval for μ (σ Unknown) (continue d) 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 Point Estimate  (Critical Value) x (Standard Error) 𝒔 𝒙 ± 𝒕 𝜶/ 𝟐 Sample S √𝒏  Student or t-Distribution  t-distribution: continuous, bell-shaped & symmetrical  Not one t-distribution: a family of t distributions  All have a mean of 0,  Their standard deviations differ according to the sample size, n.  The t-distribution is more spread out and flatter at the center than the z-distribution  Sample size increases: t-distribution approaches the z- distribution  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 x1 = 7 If the mean of these three Let x2 = 8 values is 8.0, What is x3? then x3 must be 9 (i.e., x3 is not free to vary) 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)  Student’s t Distribution Note: t z as n increases Standard Normal (t with df = ) t (df = 13) t-distributions are bell- shaped and symmetric, but have ‘fatter’ tails than the normal t (df = 5) especially for small df 0 t  Student’s t Table Upper Tail Area Let: n = 3 df.25.10.05 df = n - 1 = 2  =.10 1 1.000 3.078 6.314 /2 =.05 2 0.817 1.886 2.920 3 0.765 1.638 2.353 /2 =.05 The body of the table contains t values, 0 2.920 t not probabilities  t distribution values With comparison to the z value Confidence t t t z Level (10 d.f.) (20 d.f.) (30 d.f.) ____.80 1.372 1.325 1.310 1.28.90 1.812 1.725 1.697 1.64.95 2.228 2.086 2.042 1.96.99 3.169 2.845 2.750 2.57 Note: t z as n increases Confidence Interval Estimates for the Mean – Example The manager of the Inlet Square Mall, near Ft. Myers, Florida, wants to estimate the mean amount spent per shopping visit by customers. A sample of 20 customers reveals the following amounts spent. Based on a 95% confidence interval, do customers spend $50 on average? Do they spend $60 on average? 9-29 Confidence Interval Estimates for the Mean – Example 9-30  Approximation for Large Samples Since t approaches z as the sample size increases, an approximation is sometimes used when n  30: Technically Approximation correct for large n s s x t /2 x z /2 n n  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 z 2 σ 2  z /2 σ  n /2   e 2  e   Required Sample Size Example If  = 45, what sample size is needed to be 90% confident of being correct within ± 5? 2 2  z /2 σ   1.645(45)  n     219.19  e   5  So the required sample size is n = 220 (Always round up)  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  Confidence Intervals Confidence Intervals Population Population Mean Proportion σ Known σ Unknown  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 ( )  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 or standard error: The standard error can be Sample Distribution for estimated from the sample data:  Confidence interval endpoints Upper and lower confidence limits for the population proportion are calculated with the formula where is the standard normal value for the desired level of confidence is the sample proportion is the sample size  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  Example (continue A random sample of 100 people shows d) that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers. 1. 2. 3.  Interpretation The confidence interval for is [0.1651 ; 0.3349] 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.  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 [0.19; 0.31]  Finding the Required Sample Size for proportion problems Define the p(1  p) margin of error: e z /2 n z 2 p (1  p) Solve for n: n /2 e 2 p can be estimated with a pilot sample, if necessary (or conservatively use p =.50)  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 =.12)  What sample size...? (continue d) Solution: For 95% confidence, use Z = 1.96 e =.03 =.12, so use this to estimate p z 2 /2 p (1  p) (1.96) 2 (.12)(1 .12) n  2 450.74 e 2 (.03) So use n = 451 Confidence Interval Summary