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Prob and Stats, Nov 28 Intro to Confidence Intervals Book Sections: 8.1 Essential Questions: What is a confidence interval and how do I compute one for the mean of a population? Standards: PS-SPMJ,1, .4 Unbiased Estimators • Sample means, variances, and proportions tend to target the corresponding population parameters. • We call these 3 statistics unbiased estimators. • In other words, their sampling distributions have a mean that is equal to the mean of the corresponding population parameters. • If we want to use a sample statistic (such as a sample proportion) to estimate a population parameter (such as the population proportion), it is important that the sample statistic used as the estimator targets the population parameter. • Using a biased estimator might underestimate or overestimate the population parameter. • A good sample should be less variable and unbiased. • Variability – measured by the standard error of the mean, x ,which requires a larger sample. (Bigger sample size, smaller variation) _ UNBIASED ESTIMATORS point estimates • These target the population parameters. Mean x 2 Variance s proportion pˆ BIASED ESTIMATORS: • These do NOT target the population parameter. Median Range S tan dard Deviation s • NOTE: The bias is relatively small when using s, therefore, it is sometimes used to estimate the standard deviation of the population. Important Points • Sampling mean is unbiased since it equals the population mean. • Each individual sample may not equal the population mean, but it should be close. • When we say 95% of the data is within 2 standard deviations we mean 95% of data values are 2 . Speaking specifically about the mean….. • When speaking of samples we mean that 95% of the intervals captured will contain the population mean. Definitions • Interval Estimate – An interval bounded by two values & used to estimate the value of a population parameter. Statisticians prefer to use an interval estimate rather than a point estimate. • Level of Confidence –(1 ); the proportion of all interval estimates that involve the parameter being estimated. • Confidence Interval – An interval estimate with a specified level of confidence. What Does the Answer Mean? • If the answer of a confidence interval is (918.23, 930.23), and the confidence level was 95%, then it means that the population mean of the samples was captured 95% of the time. Remember….. • Sample measures are called statistics. • Population measures are called parameters. Using 6, 8, 12, 16, 18, 21, 22, 25, 28, 29……. • Find a point estimate for a. The mean • Answer: a. 18.5 b. Variance b. s sq. = 64.06 c. St Deviation c. s = 8.00 Confidence Interval for the Mean – Pop. Standard Deviation KNOWN Z-Interval Z Confidence Interval…… • To use, these conditions (assumptions) must be met: When: a. Sampling Distribution is normal OR b. Population standard deviation is known OR the sample size is greater than or equal to 30. ( n 30 ) Z • Z is a z-score around which the confidence interval is built. • We compute Z based on the invNorm function with 2 as its argument. Confidence Interval for Estimation of the Mean Pt . Estimate Confidence Coefficien t (St. Error of the Mean) x z ( 2 n ) This produces the lower and upper confidence levels. Definition…… • Confidence Level: the probability that the interval estimate will contain the parameter. • The most common levels of confidence are: 90%, 95%, 99% Alpha ( ) • Alpha ( ) = the total area in both tails. • Alpha/2 ( 2 ) = the total area in one tail. • Example: Confidence Level = 90% = 1 - .90 = .10 = .10 = .10/2 = 0.05 2 Z Alpha ( ) • The Z 2 = Invnorm(1- 2 ) Example • The president of a university wants to estimate the average age of students. It is known that the standard deviation is 2 years. A sample of 50 is selected and the mean age is found to be 23.2 years. Find the 95% confidence interval. What does this answer mean?...... • “We can say with 95% confidence that the average age of students at the university is between 22.65 years and 23.75 years.” Example • A certain medication increases the pulse rate. The variance is 25 beats/minute. A sample of 30 users has an average rate of 104 beats/minute. Find the 99% confidence interval. Example • A sample of 50 days showed a store served an average of 182 customers. The standard deviation was 8. Find the 90% confidence interval. How Wide are Various Confidence Intervals? • The higher the level of confidence, the wider the interval. Z / 2 Values % Confidence 90 91 92 93 94 95 96 97 98 99 Z / 2 1.64 1.70 1.75 1.81 1.88 1.96 2.05 2.17 2.33 2.58 Classwork: Handout CW 11/28/16, 1-4 Homework – Due 11/29/16, 1-2