Let's solve this step by step:

Part (a)

The test statistic $z = -2.06$.

Step 1: Find the P-value

Since this is a left-tailed test, the P-value corresponds to the area to the left of $z = -2.06$ under the standard normal distribution curve.

From a Z-table:

• For $z = -2.06$, the cumulative probability is approximately 0.0197.

Thus, the P-value = 0.019 (rounded to three decimal places).

Step 2: Compare P-value with the significance level ($\alpha = 0.10$)

• Since $P = 0.019$ is less than $0.10$, we reject the null hypothesis at the 10% significance level.

Part (b)

The test statistic $z = 1.87$.

Step 1: Find the P-value

Since this is also a left-tailed test, we look for the area to the left of $z = 1.87$.

From a Z-table:

• For $z = 1.87$, the cumulative probability is approximately 0.9693.

Thus, the P-value = 0.969 (rounded to three decimal places).

Step 2: Compare P-value with the significance level ($\alpha = 0.10$)

• Since $P = 0.969$ is greater than $0.10$, we fail to reject the null hypothesis at the 10% significance level.

Final Answers:

(a)

• The P-value is 0.019.
• At the 10% significance level, the data provide sufficient evidence to reject the null hypothesis because the obtained P-value is less than the significance level.

(b)

• The P-value is 0.969.
• At the 10% significance level, the data do not provide sufficient evidence to reject the null hypothesis because the obtained P-value is greater than the significance level.

Would you like further clarification or additional examples?

Related Questions:

1. What does the significance level mean in hypothesis testing?
2. How is a left-tailed test different from a two-tailed test?
3. What is the importance of rejecting the null hypothesis?
4. How can a Z-table be used to find P-values quickly?
5. How do we decide on a significance level in practical scenarios?

Tip:

Always interpret P-values within the context of the problem and clearly compare them to the significance level to draw appropriate conclusions.