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Chapter 6 California SAT Data There are 250,000 California High School Students We did a SRS of 500 to take the test. With a mean of X = 461 What if we had selected a different 500 students? Would the sample mean and variance be the same? Would the population mean and variance be the same? How good is our estimate of the population mean? How sure are we that the population mean is less than 500? Assume x = 100 (is this realistic?) Can we use the normal distribution? What is X ? P( X - 1.96 X < X < X + 1.96 X ) = P( X - 1.96 X < X < X + 1.96 X ) = X +/- 1.96 X is the 95% Confidence Interval for X If we took many SRS of 500, 95% of the CIs would include the true mean. We are 95% confident that the population (true) mean is within X +/- 1.96 X 95% CI is an interval calculated from sample data that is guaranteed to capture the true population parameter in 95% of all samples. Confidence Intervals gives estimates of parameters (population means) They also tell us how confident we can be that our answers are correct. They assume unbiased sampling!!!! Page 1 of 6 2 sided Confidence Level Table Value 70% 80% 90% 95% 98% 99% 99.9% 1.04 1.28 1.64 1.96 2.33 2.58 3.29 In Example 6.1 X = 461 X = 100/sqrt(500) = 4.5 X - 1.96 X = X + 1.96 X = So a 95% CI for the true population mean is ( , ) Margin of error = How big should n be to get a margin of error = 1 1.96*/sqrt(n) = 196/ sqrt(n) = 1 n= n is an integer ALWAYS ROUND UP! 99% CI for the true population mean? X - ____ X X + ____ X = = Page 2 of 6 Section 6.2 Hypothesis Testing How much evidence is there to support a claim about a population. Is there enough evidence to show a claim is true? California high school students average SAT scores higher than 485 California high school students perform better than the national average California high school students perform better than Iowa high school students Living near power lines increases your chances of getting cancer. Taking an aspirin immediately after a heart attack will increase your chances of living Hypothesis Test -- Testing the population mean ( ) 1) State a Hypothesis Ho: = o Vs H1: o Ho: o Vs H1: > o Ho: o Vs H1: < o H1 is always what you want to prove, {In this class Ho always includes = } x Z 2) Create a test statistic x 3) Compare to appropriate table value Z( / 2) {2-sided hypothesis test} Z or -Z {2-sided hypothesis test} 4) Reject or Fail to Reject Ho Page 3 of 6 Example 1) California High School students don't average 475 on there SATs Ho: = 475 Vs H1: 475 461475 311 . 2) Z 45 . 3) Since |Z| > 2.58, we are 99% confident that the average SAT score is not 475 99% 99.9% 2.58 3.29 4) We Reject Ho in favor of H1 at the 99% confidence level [ at the 99.9% Confidence level we fail to reject .001 < p-value < .01] p. 445 To test the hypothesis Ho : µ = µo at significance level , based on a SRS of size n, from a population with unknown mean µ and known , compute the test statistic 1) state the hypothesis a) H1: µ ≠ µo b) H1: µ > µo c) H1: µ < µo 2) z x x 3) find p-value a) 2*P(Z > |z| ) b) P(Z > z ) c) P(Z < z) is determined before the test is conducted p-values are based on sample data 4) Reject Ho if p-value is less than Page 4 of 6 The Survey of Study Habits and Attitudes (SSHA) is a phychological test that measures student’s study habits and attitudes toward school. Scores range from 0 to 200. The mean score for college students is about 115 and the standard deviation is about 30. A teacher suspects that the mean for for older students higher than 115. She gives the SSHA to a SRS of 25 students who are at least 30 years old. Suppose we know the scores in the population of older students are Normally distributed with standard deviation = 30. We seek evident against the claim that = 115. What is Ho and H1? If Ho is true, what is the sampling distribution? {Distribution of X } Sketch this distribution. Consider the following 2 options X =118.6 or X = 125.8, is either good evidence that the true mean is greater than 115? 1) H1: µ > 115 2) z Ho: µ 115 {Ho: µ = 115} x x 3) find p-value b) P(Z > z ) for = .05 for = .025 4) Reject Ho if p-value is less than The data must be a SRS from the population Pshyc 101 class as SRS for vision Soc 101 class as SRS to test attitudes towards the poor Different methods are needed for different designs SRS from Normal data, Shape of the population distribution and sample distribution known Page 5 of 6 Outliers can distort the result Margin of error doesn’t cover all errors How small a P is convincing? Statistical significance VS. Practical significance 6.4 Assessing performance Confidence intervals Confidence Level – how often the method succeeds (contains µ) MOE - how close (narrow) the interval is Hypothesis Tests RARELY reject Ho when it is really true [=.05 or =.01] USUALLY reject Ho when Ho is not true (i.e. Ha true) [power] Power = P(reject Ho when a specific value of Ha is true) reject Ho Ho True Ha true Type 1 error Correct! P(Type 1 error) = 1-P(Type II error) = Power =P(Correct given Ha true) fail to reject Ho Correct! Type II error P(Correct given Ho True) = 1- P(Type II error) = 1- Power sum = 1 sum = 1 Page 6 of 6