SOLUTION: State the null and alternative hypotheses, find the P - Value, find the critical value, should we reject Ho, or should we fail to reject Ho at 5% level of significance (why or w

Question 1201344: State the null and alternative hypotheses, find the P - Value, find the critical value,
should we reject Ho, or should we fail to reject Ho at 5% level of significance (why or why
not?) in the following situations:
a. The test statistic of Z = 1.00 is obtained when testing the claim that p > 0.3
b. The test statistic of Z = -2.50 is obtained when testing the claim that P < 0.75
c. The test statistic of Z = - 1.94 is obtained when testing the claim that p = 0.375

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part (a)

Null: p = 0.3
Alternative: p > 0.3

Rule: The null ALWAYS will have the equal sign.
This is to lock down the parameter we're trying to test.
The alternative will have an inequality sign of some sort (eg: "greater than")

The claim is p > 0.3 which tells us the direction of the test. We're doing a right-tailed test.
The rejection region is to the right of the critical value because of this right-tailed test.

At the 5% level of significance, the z critical value is roughly z = 1.645
This is because P(Z > 1.645) = 0.05
Approximately 5% of the area under the Z curve is to the right of 1.645.
Use a table or calculator to determine this value.

Use a Z table such as this
https://www.ztable.net/
or the table found in the back of your stats textbook

Use such a table to find that
P(Z < 1.00) = 0.84134
so,
P(Z > 1.00) = 1 - P(Z < 1.00)
P(Z > 1.00) = 1 - 0.84134
P(Z > 1.00) = 0.15866
This is the approximate p-value.

Rule: If the p-value is smaller than alpha, reject the null.

We have
p-value = 0.15866
alpha = 0.05
Therefore, we fail to reject the null.

Take note how the test statistic z = 1.00 is not to the right of the critical value z = 1.645, so we are not in the rejection region, and have further evidence to fail to reject the null.

------------------------

Summary:
Null: p = 0.3
Alternative: p > 0.3
P-value: 0.15866 (approximate)
Critical value: z = 1.645 (approximate)
Decision: fail to reject the null (aka accept the null)

=================================================

Part (b)

Null: p = 0.75
Alternative: p < 0.75
This is a left-tailed test

The critical value is now z = -1.645 because we're doing a left-tailed test
P(Z < -1.645) = 0.05 approximately

Use a Z table to find that
P(Z < -2.50) = 0.00621
which is the approximate p-value.

The p-value is smaller than alpha = 0.05, so we will reject the null. Notice the test statistic is to the left of the critical value. The test statistic is in the rejection region.

------------------------

Summary:
Null: p = 0.75
Alternative: p < 0.75
P-value: 0.00621 (approximate)
Critical value: z = -1.645 (approximate)
Decision: Reject the null

=================================================

Part (c)

Null: p = 0.375
Alternative: p =/= 0.375
The null will ALWAYS have the equal sign
In this case, the alternative hypothesis has the not equal sign (this is a two tailed test because of that).

Use a calculator or table to find the z critical values are roughly z = -1.960 and z = 1.960; a two-tailed test requires two critical values.
P(-1.960 < Z < 1.960) = 0.95 approximately
which can be rephrased as
P(Z < -1.960 or Z > 1.960) = 0.05 approximately

Rule: if the test statistic is between the critical values, then fail to reject the null. Otherwise, reject the null.

z = -1.94 is between -1.960 and 1.960

Therefore, we fail to reject the null.

Use a Z table to find that
P(Z < -1.94) = 0.02619
double this value because we're doing a two-tailed test
2*0.02619 = 0.05238

The p-value is roughly 0.05238
It is not smaller than alpha = 0.05, so this is further evidence we fail to reject the null.

------------------------

Summary:
Null: p = 0.375
Alternative: p =/= 0.375
P-value: 0.05238 (approximate)
Critical values: z = -1.960 and z = 1.960 (approximate)
Decision: Fail to reject the null

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