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HL Stats Option Confidence Testing Warm Up 3.25.15 NO CALCULATOR Hypotheses • Statistical Hypothesis – statement about the value of a population parameter. Ex. The mean or proportion • Null Hypothesis H0 – parameter takes on a definite value, meaning no effect occured. Ex. That population mean μ has value μ0 – Assumed True • Alternative Hypothesis H1 – Alternative to the Null Hypothesis. Ex. There is a difference between μ and μ0 Example Situation • Mr. Gilmartin’s roommate is testing the effect of a drug on response time by injecting 100 rats with a unit dose of the drug, subjecting each to neurological stimulus, and recording its response time. He knows that the mean response time for rats not injected with the drug is 1.2 seconds. The mean of the 100 injected rats is 1.05 with a sample standard deviation of 0.5 seconds. Do you think that the drug has an effect on response time? • Determine the Null and Alternate Hypotheses • How would you test your hypotheses? One vs. Two Tailed • One-Tailed: – H1: μ > μ0 – H1: μ < μ0 • Two Tailed – H1: μ ≠ μ0 • Could be > OR < 1 v. 2 Example • Using the example below create the Null hypothesis along with a the situation for each type of alternate hypotheses: Banana Boat is testing a new sun block and wants to compare it to its old sun block that on average provided 5 hours of protection Error Types • Type I – the mistake of rejecting the Null Hypothesis when it is in fact true • Type II – the mistake of accepting the Null Hypothesis when it is not true Normal Distribution X~N(μ,σ2) “X is Normally distributed with mean μ and variance σ2.” Z Score • • For our sample X~N(μ0,σ2/n) • Called the Null Distribution Critical Region • To reject a Null hypothesis we typically look for a 5% level of significance or lower based on the normal distribution curve we call the area that would reject the Null Hypothesis the critical region and the values the critical values Two Tailed Test • If we are looking at a two-tailed test (recall μ0 ≠ μ) we know that there is a 95% confidence interval as shown below: Two-Tailed • As a result the critical regions and values for a two-tailed test are as follows: • If our μ0 is in the critical region we reject the Null Hypothesis as our μ0 has <5% likelihood of occurring. • This also means the probability of a Type I Error is < 0.05 One-Tailed Test • See if you can create the Normal Distribution Curve and find the critical regions (at 5% significance) for both: – One-tailed (right) test – One-tailed (left) test • Explain how you found your critical values One Tailed (Right) Test • To find critical values we need P(Z≥k)=0.05 so k≈1.645 *Note – Level of significance determines the critical region and values One Tailed (Left) Test • To find critical values we need P(Z≤k)=0.05 so k≈-1.645 Critical Regions (General) p-Value Decision Making • We reject the Null Hypothesis if any of the 3 checks below hold true: 1. Test statistic is in the critical region 2. p-value is strictly less than a 3. Value is in the critical region • If we do not reject the Null then we accept it. – Note: Does not mean we have proven the Null Hypothesis, rather we don’t have enough evidence to reject it Hypothesis Testing in 7 Steps Example Check-in • Try the 6 problems on your worksheet Homework • Complete Terminology Worksheet • Part 1 and 2 of Study Guide Hypothesis Testing (Know σ) • If no effect then mean of the sample should be the same as the mean of population sample • Standard deviation of sample should = standard deviation of population / square root of sample