Confidence Intervals: Estimators
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1 Confidence Intervals: Estimators Point Estimate: a specific value at estimates a parameter e.g., best estimator of e population mean ( ) is a sample mean problem is at ere is no way to determine how close a point estimate is to e parameter Properties of a Good Estimator: 1. must be an unbiased estimator -expected value of estimator or mean obtained from samples of a given size must be equal to e parameter 2. must be consistent -as sample size increases estimator approaches value of e parameter 3. must be relatively efficient -estimator must have smallest variance of all oer estimators Interval Estimate: precise probabilities can be assigned e validity of e interval range of values at estimate a parameter e.g., mean +/ standard deviation ( ), 25 ± 10 kg, 5 to 95 -percentile
2 Confidence Intervals when is Known and n is Large Confidence Interval: interval estimate based on sample data and a given confidence level Confidence Level: probability at a parameter will fall wiin an interval estimate related to alpha ( ) level, at is, Confidence Level (CL) = 1 E.g., CL = 95% means = 0.05 CL = 99% means = 0.01 Formula for Computing Confidence Intervals where z /2 is e z-score at places e area ½ in each tail of e normal distribution. For example, if is 5% en z /2 is e z-score at places 2.5% in e right tail and 2.5% in e left tail. That is z /2 = +/
3 Margin of Error (Maximum Error of Estimate) Margin of Error also called Maximum Error of Estimate (E): Example: Compute e 95 percentile confidence interval from a sample of size 30 which has a standard deviation of 5.00 and a mean of Since is unknown use s. From z-table: z = +/ /2 Therefore, Confidence interval = CI = 25.0 ±1.789 = 23.2 to 26.8
4 Confidence Intervals when is Known and n is Large Data must be randomly sampled Each datum must be independent of oer data Data must be approximately normally distributed If not normally distributed, sample size should be greater an 30 When population standard deviation is known calculate margin of error (E) from: Add and subtract E from e sample mean Example Heights of a random sample of 100 people were collected and after determining ey were normally distributed, e mean was computed. Given at e population s standard deviation was 8.25 cm and e sample mean was cm, compute e 95 - percentile confidence interval. First, determine e z-values at define e 95 -percentile confidence interval. I.e., z = +/ Next, compute margin of error: E = (z a/2 x ) / n a/2 = (1.960 x 8.25)/ 100 = 16.17/10 = cm Finally, add and subtract E from sample mean to define e confidence interval: CI: cm < < cm
5 Confidence Intervals when is Unknown and n is Small Data must be randomly sampled and independent. If data are normally distributed and is not known, which is often e case, use sample standard deviation, s, and use e t-distribution wi n 1degrees of freedom. If not normally distributed use t-distribution as long as n > 30. Formula for Computing Confidence Intervals for Small Sample Sizes Decision Tree for Selecting Statistical Meod
6 t-distribution family of curves similar to z-distribution which become more platykurtic (flatter) as sample size decreases select distribution using degrees of freedom (df) at is usually n-1 mean is 0, area under curve is 1, is asymptotic to x-axis, symmetrical about mean, SD is greater an 1, approaches shape of z-distribution as sample size increases (very similar when df =29 or greater. Example: Compute e 95 -percentile confidence interval from a sample of size of 10 which has a standard deviation of 5.00 and a mean of (Similar to previous example.) From t-table wi degrees of freedom (df) = n 1 = 9: t /2 = +/ (Use columns labelled area in two tails or two tails.) Therefore, Confidence interval = CI = 25.0 ± 3.58 = 21.4 to 28.6
7 Sample Size Estimation Minimum Sample Size for Interval Estimate of Population Mean: where n is sample size, is e population standard deviation and E is e maximum error of estimate. Note, always round up to e next highest integer when ere is a fraction. When is unknown it may be estimated from e sample standard deviation, s. Example: Calculate e sample size needed to estimate muscle streng from a population at has a standard deviation of newtons if you want to be 95% confident and wiin 50.0 newtons. 2 n = ( / 50.0) = 30.1 You will need a sample size of 31.
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