Hypothesis Testing The z-test for Proportions

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$Math 110 Test # 1$

$Calculus 2 for Engineers (Math 124) - Department of Mathematics ...$

• If you are trying to prove a difference H A : p ≠ p • This is the alternative Hypothesis (H A) - the Research Hypothesis Example • A new surgical procedure is developed for correcting heart defects infants before the age of one month. • Previously the procedure was used on infants that were older than one month and the success rate was 91% • A study is conducted to determine if the success rate of the new procedure is greater than 91% (n = 200) 0 Comments • Different objectives will result in different choices of the alternative hypothesis • If you were interested in Proving that the new procedure is an improvement: • Then H A : p > p 0 • If you were interested in proving that the new procedure is not an improvement: • Then H A : p Then H A : p ≠ p We want to test – H0 : p ≤ 0. 91( 91%) Against p > 0. 91( 91 – H A : %) p = the success rate of the new procedure 0 0 4
Performing the Test 1. Decide on α = P[Type I Error] = the significance level of the test Choose (α = 0.05) 2. Collect the data • The number of successful operations in the sample of 200 cases is x = 187 x 187 pˆ = = = n 200 3. Compute the test statistic 0. 935 (93.5%) pˆ − p0 z = = σ pˆ pˆ − p0 p0( 1− p0 ) n = 0. 935− 0. 91 = 1. 235 0. 91( 1− 0. 91) 200 4. Make the Decision zα = z • Accept H0 if: z ≤1. 645 • Reject H 0 if: z > 1. 645 0 . 05 = 1. 645 Since the test statistic is in the Acceptance region we decide to Accept H 0 Conclude that H0 : p ≤ 0. 91( 91%) is true There is a no significant (α = 5%) increase in the success rate of the new procedure over the older procedure Comments • When the decision is made to accept H 0 is made it should not be conclude that we have proven H 0 . • This is because when setting up the test we have not controlled β = P[type II error] = P[accepting H 0 when H 0 is FALSE] • Whenever H 0 is accepted there is a possibility that a type II error has been made. In the last example The conclusion that there is a no significant (α = 5%) increase in the success rate of the new procedure over the older procedure should be interpreted: We have been unable to proof that the new procedure is better than the old procedure An analogy – a jury trial The two possible decisions are – Declare the accused innocent. – Declare the accused guilty. 5
Page 1 and 2: Hypothesis Testing To define a stat
Page 3: α The Test Statistic pˆ − p z =
Page 7 and 8: The Data • Let x1 , x2 , x3 , …
Page 9: Test Statistic z = = x − μ0 x

Hypothesis Testing The z-test for Proportions

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