Questions: (a) Find the P - value for the test statistic z=-1.41 for the following null and alternative hypotheses: H0: The population mean is 50. Ha: The population mean is less than 50. The P - value is

(a) Find the P - value for the test statistic z=-1.41 for the following null and alternative hypotheses:
H0: The population mean is 50.
Ha: The population mean is less than 50.
The P - value is

Transcript text: (a) Find the $P$ - value for the test statistic $z=-1.41$ for the following null and alternative hypotheses: $H_{0}$ : The population mean is 50 . $H_{a}$ : The population mean is less than 50. The $P$ - value is

Solution

Solution Steps

Step 1: Define the Hypotheses

We are testing the following hypotheses:

Null Hypothesis $ H_0 $: The population mean is $ \mu = 50 $.
Alternative Hypothesis $ H_a $: The population mean is less than $ 50 $.

Step 2: Calculate the Z-score Range

For the given test statistic $ z = -1.41 $, we define the Z-score range for the left-tailed test:

Lower bound: $ -\infty $
Upper bound: $ -1.41 $

Step 3: Calculate the P-value

The P-value is calculated as follows: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-1.41) - \Phi(-\infty) \] From the calculations, we find: \[ P = 0.0793 \]