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For a test of $ho$: $p$ = $0.5$, the $z$ test statistic equals $1.74$. Find the $z$ test statistic equals $p$-value for $ha$: $p$ > $0.5$.

The purpose of this question is to find the $p$ value for $H_a$: $p$ > $0.5$ using the z-test statistics test.

The $p$-value approach determines the “likely or unlikely” probability, assuming the null hypothesis is true and determining more extreme test statistics in the direction of the alternative hypothesis than the one observed.

If the $p$-value is less than acceptable probability, then the probability is “unlikely” and the null hypothesis is rejected. If the $p$-value is greater than the acceptable probability, then the probability is “likely” and  the null hypothesis is acceptable.

Expert Answer:

The value of $p$ can be calculated by using a standard normal table:

This standard normal table gives the value of z-test statistics, which is:

\[z= 1.74\]

Null hypothesis is represented by $H_o$. The given value of $p$ for $H_o$  test is:

\[p= 0.5\]

We have to find the $z$-test statistics value when the $p$-value for the alternating hypothesis is given as:

\[H_a: p>0.5\]

The value of $z$ statistics tells us the kind of test. For example, in this question, if the value of $z$ statistics is $-1.74$, then the test is a left tailed test. On the other hand, if the value of $z$ is $1.74$, then the test is called the right tailed test.

Numerical Results:

The formula for right tail test is given as:

\[p = 1 – P (Z > z)\]

By putting given values:

\[p = 1- P (Z > 1.74)\]

The standard probability table will be used to find the value of $p$ at $1.74$ .

By putting values from the table, we get:

\[p = 1- 0.9591\]

\[p = 0.0409\]

The value of $p$ is $0.0409$ for $H_a$: $p$ > $0.5$.

Alternate Solution:

We can also find the value of $p$ by simply looking at the standard probability table. It includes two steps:

Step 1: Look out for $1.74$ from the rows.

Step 2: Look out for the value of $p$ against the row of $1.74$.

The value of $p$ will be $0.0409$.

Example:

If the $p$-value is less than 0.5, that is $p$ < $0.5$, then we will take the left tailed test where the null hypothesis is given as:

\[H_o: p = 0.5\]

The value of $p$ for alternating hypothesis is:

\[H_a: p < 0.5\]

The formula for $p$-value is given as:

\[p = P ( Z < z)\]

By putting values in the above formula:

\[p = P ( Z < -1.74)\]

by using standard probability table to find the $p$-value at $-1.74$:

\[p = 0.0409\]

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