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Chapter 8: Q. 9.70 (page 370)

The symbol (z) is often used to denote the area under the standard normal curve that lies to the left of a specified value of z. Consider a one-mean z-test. Denote z0 as the observed value of the test statistic z. Express the P-value of the hypothesis test in terms of if the test is

a. left tailed.

b. right tailed.

c. two tailed.

Short Answer

Expert verified

(a)∅(z0)

(b) 1-∅(z0)

(c) 21-P(z-z0)2∅(-z0)

Step by step solution

01

Step 1. Given

Consider a one-mean z-test. Denote z0 as the observed value of the test statistic z.

02

Step 2.  The test statistics for a mean test.

The test statistic for a one-mean z-test with null hypothesis given by H0:μ = μ0 is

z=x-μ0σ-n

If the null hypothesis is true the test statistic has the standard normal distribution i.e.

=zN(0,1).

03

Part(a) Step 3. Calculation

Left-failedtestP-value=p(z≤z0),wherez~N(0,1)=∅(z0)

04

Part(b) Step 4. Calculation

Right-failedtestP-value=p(z≤z0),wherez~N(0,1)=1-∅(z0)

05

Part(c) Step 5. Calculation

Two-failedtestP-value=P(z≥z0)+P(z≤-z0),wherez~N(0,1)=P(z≤-z0)+P(z≤-z0)P(z≥z0)+P(z≥z0)StandardNormalcurveissymmetricaboutis0P(z≤-z0)=P(z≥z0)2P(z≤-z0)2P(z≥z0)...(**)21-P(z-z0)2∅(-z0)

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