95% of the samples of size 500 will capture µ between x 9

Transcription

1 Wednesday Chapter 10 Introduction to inference Statistical inference provides methods for drawing conclusions about a population from sample data Estimating with confidence SAT σ = 100 n = 500 µ = 461 For sample (σ x = σ/ ) σ x = 100/ = 4.5 µ - 9 µ µ % 95% of the samples of size 500 will capture µ between x = = % between 452 and 470

2 We are 95% confident that the true mean of the SAT for California falls between 452 and 470. Margin of error how accurate we believe our guess is. Confidence interval A level C confidence interval for a parameter has two parts 1. An interval calculated from the data, usually of the form Estimate margin of error x 2 standard deviations 2. A confidence level, C, which gives the probability that the interval will capture the true parameter value in repeated samples. C.9 or 90 % Page 541 picture (need to look at) Homework read pages do problems 1-4

3 Thursday Confidence interval for a population mean Conditions for constructing a confidence interval for µ 1. Data comes from SRS of the population of interest 2. Sampling distribution of x is approximately normal _ _ Need to know Confidence Tail area Z* 80% % % %

4 Can find the z* at the bottom of the table in the back of book. Z* is called critical value (on the handout z table Z* is noted as ) Critical values The number z* with probability p lying to its right under the standard normal curve is called the upper p critical value of the standard normal distribution Probability p Z* Confidence interval for a population mean Choose an SRS of size n from population having unknown µ and known σ. A level C confidence interval for µ is X z* ( ) Where z* is the value with an area C between z* and z* under the standard normal curve.

5 Ex page Confidence intervals 1. Identify population of interest and the parameter 2. Choose the appropriate inference procedure. Verify the conditions for using the procedure. 3. If conditions are met, do procedure CI = estimate margin of error 4. Interpret results ---Context!!!!! Margin of error gets smaller when Z* gets smaller σ gets smaller n gets larger choosing sample size m = z*( ) 95% CI m 5 σ = 43 5 (1.96) (43/ ) 5 (1.96) (43) N n 285

6 Cautions page 553 Homework 8-10, 20, 21, 24, 25 Friday Worksheet quiz + problems 1,2, and 6 Wednesday 10.2 tests of significance Assess the evidence provided by data about some claim concerning a population Hypothesis testing Null Hypothesis states there is no change or effect on the population Alternate Hypothesis there is a change P value µ Probability of a result at least as far Out as the result we actually got

7 Statistically significant a result with a p-value less than.05 (Chance alone would rarely produce so extreme a result) Outline of a test 1. Describe µ 2. Calculate x that estimates the parameter (if value is far enough out reject the null) 3. P-value says how unlikely a result is 4. State Hypothesis a. H o null hypothesis (THIS IS ALWAYS EQUAL) No effect we are testing the strength of evidence against the null) b. H a alternate hypothesis H a : µ One sided H a : µ One sided H a : µ Two sided Read problems Thursday p-value p-value Smaller p-value the more extreme Better chance of rejecting null

8 The observations are an SRS of size 10 for a normal population with σ = 1. The observed mean sweetness loss for one cola was x= 0.3. The p- value for testing H o : µ= 0 H a : µ 0 EX. SRS n= 10 x =.3 σ = 1 P(x.3) σ x = 1/ =.316 z = = = Statistically significant if p-value is as small or smaller than α We say it is statistically significant at the level α α =.05 unless otherwise stated Significance test 1. Identify population of interest, state null and alternate hypothesis in words and in symbols 2. Choose appropriate inference procedure and verify conditions 3. If conditions are met carry out procedure Calculate test procedure Find p value 4. Interpret results context of the problem

9 Z-test for a population mean H o : µ = µ o SRS size n Population unknown µ Standard deviation σ One sample Z-statistic Z= H a : µ µ o H a : µ < µ o H a : µ o Ex: H o : µ = 128 Comparing exec s blood pressure is 128 H a : µ 128 Comparing exec's blood pressure is different than 128 (2-sided test) σ = 15 n = 72 x =

10 Z= = sided test find area then multiply by two 2(.1379) =.2758 More than 27% of the time, an SRS of size 72 from the general male population would have a mean blood pressure at least as far from 128 as that of the executive sample. The observed x = is therefore not good evidence that executives differ from other men. We fail to reject our null hypothesis. Or you can use a confidence level to reject a hypothesis Confidence level X X_ If falls outside of the range than reject H o Homework read pages do problems 39 and 43

11 Friday 10.3 Making sense of statistical significance Choosing a level of significance Choose α by asking how much evidence is required to reject H o. This depends on two circumstances 1. How plausible is H o? If H o represents an assumption that people have believed for years, strong evidence, (small α) will be needed. 2. What are the consequences of rejecting? Go over ex page 588 Go over page 590 Do problems 3-5 from worksheet plus problems 40,44,46,47 Quiz Monday on sections 10.1 and 10.2 Monday Quiz And do problems pages

12 Tuesday 10.4 Inference as a decision Type I error: we reject H o when it is really true Type II error: we accept H o when in fact it is false