Question

Explain how the P -value is obtained for a mean z-test in case of hypothesis test is

(a) left-tailed (b) right- tailed (c) two tailed

Short answer

(a)Area under standard normal curve that lies to the left of z_0.

(b)Area under standard normal curve that lies to the right of z0

(c)Area under standard normal curve that lies outside the interval

Step-by-step solution

Step 1. Given

Z-case hypothesis test

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

$z=\frac{\overline{x}-{\mu }_0}{\sigma -\sqrt{n}}$

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

=zN(0,1).

Part (a) Left-tailed test

$P-value=P(z\le z_0),wherez~N(0,1)$

The P-value of a left-tailed test is the chance of seeing a value of the test statistic z that is equal to or less than the value actually seen.

To the left of z0, the area under the standard normal curve.

Part( c) Right tailed test

The P-value of a right-tailed test is the chance of seeing a value of the test statistic z that is equal to or greater than the value actually seen.

$P-value=P(z\ge z_0),wherez~N(0,1)$

Area under standard normal curve that lies to the right of z₀

Part( c) Two -tailed test

Two-tailed test: The P-value is the likelihood of seeing a value of the test statistic z that is at least twice as large as the actual value.

$i.e.P-value=P(z\le \left|-z_o\right|)+P(z\ge \left|z_0\right|),wherez~N(0,1).......(*)$

Area under standard normal curve that lies outside the interval from $-\left|z_0\right|to\left|z_0\right|$

Because the normal curve is symmetric about 0,

The P-value of a two-tailed one mean z test can be stated as follows using the relation () in equation ():

$P-value=\left\{\begin{array}{l}2P(z\ge z_0)\\ 2P(z\le z_0)\end{array}\right.$