Statistics 300: Elementary Statistics Sections 7-2, 7-3, 7-4, 7-5 - PDF document

Statistics 300: Elementary Statistics Sections 7-2, 7-3, 7-4, 7-5 Parameter Estimation • Point Estimate – Best single value to use • Question – What is the probability this estimate is the correct value? Parameter Estimation • Question – What is the probability this estimate is the correct value? • Answer – zero : assuming “ x” is a continuous random variable – Example for Uniform Distribution 1

If X ~ U[100,500] then • P(x = 300) = (300-300)/(500-100) • = 0 100 300 400 500 Parameter Estimation µ • Pop. mean – Sample mean x p • Pop. proportion ˆ – Sample proportion p σ • Pop. standard deviation – Sample standard deviation s Problem with Point Estimates • The unknown parameter ( µ , p, etc.) is not exactly equal to our sample-based point estimate. • So, how far away might it be? • An interval estimate answers this question. 2

Confidence Interval • A range of values that contains the true value of the population parameter with a ... • Specified “level of confidence”. • [L(ower limit),U(pper limit)] Terminology • Confidence Level (a.k.a. Degree of Confidence) – expressed as a percent (%) • Critical Values (a.k.a. Confidence Coefficients) Terminology • “alpha” “ α ” = 1-Confidence – more about α in Chapter 7 • Critical values – express the confidence level 3

Confidence Interval for µ lf σ is known (this is a rare situation) = ± x E  σ  = ⋅   E z α 2   n Confidence Interval for µ lf σ is known (this is a rare situation) if x ~N(?, σ ) σ   = ± ⋅   x z α   n 2 Why does the Confidence Interval for µ look like this ? σ   = ± ⋅   x z α   n 2 4

σ µ x ~ N ( , ) n make an x value into a z - score. The general z - score expression is − µ ) = x ( z σ for x , µ µ is : unchanged x and σ σ is x n 5

so a z - score based on x is − µ x = z σ n Using the Empirical Rule Make a probabilit y statement :     − µ ( x ) − < < =   P 2 2 95 % σ     n Normal Distribution 0 4 . 5 Relative likelihood 0 . 4 0 α 0 . 3 5 α         0 . 3 0     2 2 0 2 . 5 . 0 2 0 0 1 . 5 0 . 1 0 0 . 5 0 . 0 0 0 -3 -2 -1 0 1 2 3 Value of Observation 6

Check out the “Confidence z-scores” on the WEB page. (In pdf format.) Use basic rules of algebra to rearrange the parts of this z-score. Manipulate the probabilit y statement :       ( ) σ σ − < − µ < =       P 2 x 2 0 . 95       n n 7

Manipulate the probabilit y statement :       σ σ − − < − µ < − + =       P x 2 x 2 0 . 95       n n Confidence = 95% α = 1 - 95% = 5% α/2 = 2.5% = 0.025 Manipulate the probabilit y statement : multiply t hrough by (-1) and change the order of the terms       σ σ  − < µ < +  =     P x 2 x 2 0 . 95       n n Confidence = 95% α = 1 - 95% = 5% α/2 = 2.5% = 0.025 Confidence Interval for µ lf σ is not known (usual situation)   s = ± ⋅   x t α   n 2 8

Sample Size Needed to Estimate µ within E, with Confidence = 1- α 2   ⋅ σ ˆ Z α   = 2 n   E   Components of Sample Size Formula when Estimating µ • Z α /2 reflects confidence level – standard normal distribution σ σ • ˆ is an estimate of , the standard deviation of the pop. • E is the acceptable “margin of error” when estimating µ Confidence Interval for p • The Binomial Distribution gives us a starting point for determining the distribution of the sample proportion : p ˆ x successes = = ˆ p n trials 9

For Binomial “x” µ = np σ = npq For the Sample Proportion 1 ( ) x = = ˆ p x n n x is a random variable n is a constant Time Out for a Principle: µ If is the mean of X and “a” is a constant, what is the mean of aX? ⋅ µ . a Answer: 10

Apply that Principle! • Let “a” be equal to “1/n”   1 X • so = = =   p ˆ aX X   n n µ = µ = • and a a ( np ) ˆ p x   1 = ⋅ =   np p   n Time Out for another Principle: σ 2 If is the variance of X and “a” x is a constant, what is the variance of aX? σ = a σ . 2 2 2 Answer: aX x Apply that Principle! • Let x be the binomial “x” • Its variance is npq = np(1-p), which is the square of is standard deviation 11

Apply that Principle! • Let “a” be equal to “1/n”   1 X = = =   • so p ˆ aX X   n n ( ) σ = σ = 2 2 2 2 • and a 1 / n ( npq ) ˆ p X Apply that Principle! 2   1 pq ⋅ = = σ   2 npq p ˆ   n n and pq σ = ˆ p n When n is Large,   pq   µ = σ = ˆ p ~ N p ,    n  12

What is a Large “n” in this situation? • Large enough so np > 5 • Large enough so n(1-p) > 5 • Examples: – (100)(0.04) = 4 (too small) – (1000)(0.01) = 10 (big enough) Now make a z-score − = ˆ p p z pq n And rearrange for a CI(p) Using the Empirical Rule Make a probability statement:     − ˆ p p   − < < = P 1.96 1.96 95%   pq     n 13

Normal Distribution 0 4 . 5 Relative likelihood . 0 4 0 0 . 3 5  α   α      0 . 3 0     2 2 0 2 . 5 0 . 2 0 0 1 . 5 0 . 1 0 0 . 0 5 . 0 0 0 -3 -2 -1 0 1 2 3 Value of Observation Use basic rules of algebra to rearrange the parts of this z-score. Manipulate the probability statement: pq Step 1: Multiply through by : n   pq pq ( ) − < − < =   P 1.96 p ˆ p 1.96 0.95    n n  14

Manipulate the probability statement: Step 2: Subract from all parts of the expression: ˆ p   pq pq − − < − < − + =   P p ˆ 1.96 p p ˆ 1.96 0.95    n n  Manipulate the probability statement: Step 3: Multiply through by -1: (remember to switch the directions of < >)   pq pq + > > − =   ˆ ˆ P p 1.96 p p 1.96 0.95    n n  Manipulate the probability statement: Step 4: Swap the left and right sides to put in conventional < < form: p   pq pq − < < + =   ˆ ˆ P p 1.96 p p 1.96 0.95   n n   15

Confidence Interval for p (but the unknown p is in the formula. What can we do?) pq = ± ⋅ ˆ p z α n 2 Confidence Interval for p (substitute sample statistic for p) p ˆ q ˆ = ± 2 ⋅ ˆ p z α n Sample Size Needed to Estimate “p” within E, with Confid.=1- α   2 Z   α 2 ⋅ = ˆ ˆ n p q     2 E   16

Components of Sample Size Formula when Estimating “p” • Z α /2 is based on α using the standard normal distribution • p and q are estimates of the population proportions of “successes” and “failures” • E is the acceptable “margin of error” when estimating µ Components of Sample Size Formula when Estimating “p” • p and q are estimates of the population proportions of “successes” and “failures” • Use relevant information to estimate p and q if available • Otherwise, use p = q = 0.5, so the product pq = 0.25 Confidence Interval for σ starts with this fact µ σ if ~ ( , ) x N then ( ) n − 2 1 s χ 2 ~ (chi square) σ 2 17

What have we studied already that connects with Chi-square random values? ( ) n − 2 1 s χ 2 ~ (chi square) σ 2 ∑ ( ) − µ 2 x ( ) − n 1 ( ) ( ) − − 2 n 1 s n 1 = σ σ 2 2 ∑ ( ) − µ 2 x = σ 2 ( ) ( )   2 − µ 2  − µ  x x ∑ ∑   =   σ σ 2       ∑ = 2 a sum of squared z standard normal values 18

Confidence Interval for σ ( ) − 2 n 1 s = LB χ 2 R ( ) − 2 n 1 s = UB χ 2 L 19