Parameter Estimation II

Transcription

1 Parameter Estimation II ELEC 41 PROF. SIRIPONG POTISUK Estimating μ With Unnown σ This is often true in practice. When the sample is large and σ is unnown, the sampling distribution is approimately normal regardless of the underlying distribution When the sample is small and σ is unnown, we must mae an assumption about the form of the underlying distribution to obtain a valid CI procedure. A reasonable assumption in many cases is that the underlying distribution is normal. 1

2 A Large Sample CI for μ (Unnown σ) CI on μ of a Normal Distribution (Unnown σ) Let X and S be the sample mean and variance of a random sample from a normally distributed population with unnown μ and σ. The random variable T = X S / μ n has a Student-t distribution with n - 1 degrees of freedom.

3 3 Student-t Distribution df and 1 1 ) ( = < < + Γ = f df and, 1 ) ( 1 = < < + Γ = + f X π, } { 0 } { > = = X Var X E The t Confidence Interval on μ

4 Which (if any) distribution to use? Procedure for the interval estimation of μ with σ unnown 4

5 Eample A manager of a paint store, wants to estimate the mean amount of a product sold per day. Twenty business days are monitored, and an average of 3 gallons is sold daily. The sample standard deviation is 1 gallons. Calculate the confidence limits at the 95% confidence level. 5

6 Eample A random sample has been taen from a normal population with the following statistics: N Mean SEmean Stdev Variance Sum 10? 0.507?? (a) Find the missing quantities (?) (b) Construct a 95% CI on the population mean 6

7 Eample The compressive strength of concrete is being tested by a civil engineer. Twelve specimens are tested and the following data are obtained: (a) Chec the assumption that compressive strength is normally distributed. (b) Construct a 95% two-sided confidence interval on the mean strength. 7

8 Estimating the Population Variance Variance shows the etent of the spread or scatter in a data set It is desirable to now such variation so that steps can be taen to control it Tire manufacturer wants to be sure that tires produced are of consistent mileage quality A drug company must focus on the potency of the tablets so that some are not unduly wea while others do not produce overdoses Estimating the Population Variance Let S be the sample variance of a random sample taen from a normally distributed population, the sampling distribution of the sample variance follows a chi-square distribution, i.e., the RV X ( n 1) = S σ has a chi-square (χ ) distribution with n -1 degrees of freedom. 8

9 9 Chi-Square Distribution df and 0, 1 ) ( 1 = > Γ = e f X X V X E } { } { = X Var } { = Chi-Square Distribution

10 CI on the Variance of a Normal Population Eample: A Pressing Problem The strength and conditioning coach of the Citadel football team described the outcomes of a weight- lifting and fitness program he designed. As part of the program s evaluation, he had each player do a one-repetition, maimum-weight bench press. The weights pressed by the linebacers are: 340, 380, 305, 335, 375, 400, 305, 385, and 315. Construct t a 90% confidence interval for the standard deviation in the maimum weights pressed by the population of linebacers who go through this program. 10

11 Estimating Population Proportions Interested in estimating population proportions (or percentages) when dealing with attribute (categorical) data Percent of defective items produced by a machine Percent of minority students at US colleges Estimate a population proportion or percentage on the basis of sample results 11

12 Sampling Distribution of Proportions Population proportion denoted by p Sample proportion denoted by pˆ p and defined as pˆ = ( /n) 100, where is the number of items in a random sample possessing the characteristic of interest, and n is the sample size pˆ p is a point estimate of p The underlying population distribution is the Binomial distribution with mean np and np (1-p) variance Sampling Distribution of Proportions The sampling distribution of proportions P is a distribution of the proportions of all possible samples that could be taen in a given situation, where the samples are simple random samples of fied size n. μ p and σ p are the mean and the standard deviation (standard error) of the distribution, respectively Pˆ 1

13 Normal Approimation for Binomial Proportion Based on the Central Limit Theorem If p is not too close to 0 or 1 and n is relatively large (i.e., np and n(1- p) 5, the sampling distribution of proportions Pˆ is approimately normal. That is, Pˆ is approimately normal with μ p = p and σ p = p( 1 p) n Eample: Did They Inhale? In 1996, it is estimated that 33.1% of all college students used marijuana in the previous 1 months. Suppose a random sample of 80 such students is taen and assume that the 33.1% is the actual percentage. What is the probability that the percentage in this sample who have used marijuana is over 30%? 13

14 Confidence Interval For Population Proportion Procedure for the interval estimation of p using large samples 14

15 Eample: Reelection Bid Political Polls represent one of the major uses of interval estimation of p. Let s assume that President Barac Obama faces a tough reelection campaign and orders a poll to learn how the voters view his candidacy. A random sample of 100 voters reveals that 53 are liely to vote for him, while the others prefer his opponent or undecided. At the 95 percent level of confidence, what s the population percentage of voters who epress a preference for him? 15

16 Choice of sample size If pˆ is used as an estimate of p, we can be 100(1- α)% confident that the error will not eceed a specified amount E when the sample size is zα / n = p(1 p) E An upper bound on n is given by n = z α / E (0.5) Eample Suppose you have been ased by the Red Cross to estimate the percentage of cadets & non-cadets who are willing to donate a pint of blood. The estimate should be within ± 5 percent of the true percentage with a confidence level of 95 %. How big should the sample size be? Assume you have no idea of the true percentage. 16