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For a test of Ho: p=0.5,the z test statistic equals -1.74. Find the p-value for Ha: p<0.5.

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.

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