The question aims to find out the p-value using the given alternative hypothesis, which is a one-sided hypothesis. Therefore, the p-value will be determined for the left tail test with reference to the standard normal probability table.
When the alternative hypothesis states that a certain value for a parameter in the null hypothesis is lesser than the actual value, then left-tail tests are used.
Figure-1 : P-Value and Satistical Significance
Let’s first understand the difference between the Null and Alternative hypotheses.
Null hypothesis $H_o$ refers to no association between two parameters of the population, meaning both are the same. Alternative hypothesis $H_a$ is opposite to the null hypothesis and states that there is a difference between two parameters.
Expert Solution:
In order to calculate the p-value, we will use the standard normal table.
According to the given information, the value of the test statistic is given as:
\[ z = -1.74 \]
Null hypothesis $H_o$ is given as:
\[ p = 0.5 \]
Alternative Hypothesis $H_a$ is given as:
\[ p < 0.5 \]
The formula for p-value is given as:
\[ p = P (Z < z) \]
Where P is the probability:
\[ p = P (Z < -1.74) \]
The p-value can be calculated by determining the probability less than -1.74 using the standard normal table.
Therefore, from the table p-value is given as:
\[ p = 0.0409 \]
Alternative Solution:
For the given problem, the p-value will be determined using the standard probability table. Check against the row starting with -1.74 and column with 0.04. The answer obtained will be:
\[ p = P ( Z< -1.74) \]
\[ p = 0.0409 \]
Therefore, the p-value for $H_a$ < 0.5 is 0.0409.
Example:
For a test of $H_o$: \[ p = 0.5 \], the $z$ test statistic equals 1.74. Find the p-value for
\[ H_a: p>0.5 \].
Figure-2 : Z-Test Satistic
In this example, the value of test statistic $z$ is 1.74, therefore, it is a right tail test.
For calculating the p-value for a right tail test, the formula is given as:
\[ p = 1 – P ( Z > z) \]
\[ p = 1 – P ( Z > 1.74) \]
Now use the standard probability table to find the value.
The p-value is given as:
\[ p = 1 – 0.9591 \]
\[ p = 0.0409 \]
Therefore, the p-value is 0.0409.