Hypothesis Testing The z-test for Proportions

More documents

Recommendations

Info

$Math 110 Test # 1$

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

The null hypothesis (H 0 ) – the accused is innocent The alternative hypothesis (H A ) – the accused is guilty The two possible errors that can be made: – Declaring an innocent person guilty. (type I error) – Declaring a guilty person innocent. (type II error) Note: in this case one type of error may be considered more serious Requiring all 12 jurors to support a guilty verdict : – Ensures that the probability of a type I error (Declaring an innocent person guilty) is small. – However the probability of a type II error (Declaring an guilty person innocent) could be large. Hence: When decision of innocence is made: – It is not concluded that innocence has been proven but that – we have been unable to disprove innocence The z-test for the Mean of a Normal Population We want to test, μ, denote the mean of a normal population Situation • A success-failure experiment has been repeated n times • The probability of success p is unknown. We want to test – H0 : p = p0 (some specified value of p) Against – HA : 0 p p ≠ 6
The Data • Let x1 , x2 , x3 , … , xn denote a sample from a normal population with mean μ and standard deviation σ. • Let n ∑ xi i= 1 x = = the sample mean n • we want to test if the mean, μ, is equal to some given value μ 0 . • Obviously if the sample mean is close to μ 0 the Null Hypothesis should be accepted otherwise the null Hypothesis should be rejected. The Test Statistic • To decide to accept or reject the Null Hypothesis (H 0 ) we will use the test statistic x − μ0 x − μ0 z = = = σ σ x x − μ0 n ≈ σ x − μ0 n s n • If H0 is true we should expect the test statistic z to be close to zero. • If H 0 is true we should expect the test statistic z to have a standard normal distribution. • If H A is true we should expect the test statistic z to be different from zero. The sampling distribution of z when H 0 is true: The Standard Normal distribution Reject H 0 0 z Accept H 0 Reject H 0 P P Reject H 0 α/2 The Acceptance region: [ Accept H 0 when true] = P[ − zα / 2 ≤ z ≤ zα / 2] = 1− α [ Reject H when true] P[ z < −z or z > z ] = α 0 α/2 0 z − zα / 2 zα / 2 Accept H 0 • Acceptance Region – Accept H0 if: − z ≤ z ≤ z Reject H 0 = α / 2 α / 2 α / 2 α / 2 • Critical Region – Reject H0 if: z < −z z > z • With this Choice α / 2 or α / 2 [ I Error] [ Reject when true] H P P = Type 0 [ < − > ] = α z z z z = α / 2 α / 2 or P Summary To test mean of a Normal population H0 : μ = μ0 (some specified value of μ) Against HA : μ ≠ μ0 1. Decide on α = P[Type I Error] = the significance level of the test (usual choices 0.05 or 0.01) 7
Page 1 and 2: Hypothesis Testing To define a stat
Page 3 and 4: α The Test Statistic pˆ − p z =
Page 5: Performing the Test 1. Decide on α
Page 9: Test Statistic z = = x − μ0 x

Hypothesis Testing The z-test for Proportions

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?