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Chapter 8: Q. 9.69 (page 370)

Consider a one-mean z-test. Denote z0 as the observed value of the test statistic z. If the test is right tailed, then the P-value can be expressed as P(z≥z0). Determine the corresponding expression for the P-value if the test is

a. Left failed b. Two-failed

Short Answer

Expert verified

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

(b) Therefore, one can compute the P-value of a two tailed z-test by using the expression given in (*) or by using anyone of the two expressions given in (**)

Step by step solution

01

Step 1. Given

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.

=z~ N(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

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)Normalcurveissymmetricaboutis0P(z≤-z0)=P(z≥z0)2P(z≤-z0)2P(z≥z0)...(**)

Therefore, one can compute the P-value of a two tailed z-test by using the expression given in (*) or by using anyone of the two expressions given in (**)

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