91Ó°ÊÓ

Four different hypothesis testing problems are provided. The values of the test statistics and significance level for the respective hypothesis are given.

a.

The provided scenario is the one-sided right-tailed test.

The p-value for the right-tailed test is:

\(\begin{aligned}p &= P\left( {Z > 2.02} \right)\\ &= 1 - P\left( {Z \le 2.02} \right)\\ &= 1 - 0.9783\\ &= 0.0217\end{aligned}\).

The z-table is used to obtain the probability of a z-score less than or equal to 2.02.

The p-value is greater than 0.01; therefore, do not reject the null hypothesis.

b.

The p-value for the left-tailed test is:

\(\begin{aligned}p &= P\left( {Z < - 1.78} \right)\\ &= 0.0375\end{aligned}\).

The value at the intersection of -1.70 and 0.08 is the required probability in the z-table.

The p-value is less than 0.05; therefore, reject the null hypothesis.

c.

The p-value for the two-tailed test is:

\(\begin{aligned}p &= 2P\left( {Z < - 1.93} \right)\\ &= 2 \times 0.0268\\ &= 0.0536\end{aligned}\).

The value at the intersection of -1.90 and 0.03 is the required probability in the z-table.

The p-value is less than 0.1; therefore, reject the null hypothesis.

d.

The p-value for the two-tailed test is:

\(\begin{aligned}p &= 2P\left( {Z > 1.96} \right)\\ &= 2\left( {1 - P\left( {Z \le 1.96} \right)} \right)\\ &= 2\left( {1 - 0.9750} \right)\\ &= 2 \times 0.0250\\ &= 0.05\end{aligned}\).

The z-table is used to obtain the probability of a z-score less than or equal to 1.96.

The p-value equals the significance level \(\alpha = 0.05\); therefore, do not reject the null hypothesis.