Question 1168838
Absolutely, let's break down this hypothesis test step-by-step.

**1. Define the Hypotheses**

* **Null Hypothesis (H₀):** The proportion of wrong test results is 10% or more (p ≥ 0.10).
* **Alternative Hypothesis (H₁):** The proportion of wrong test results is less than 10% (p < 0.10).

This is a left-tailed test.

**2. Set the Significance Level**

* α = 0.01

**3. Calculate the Sample Proportion**

* Sample size (n) = 311
* Number of wrong results (x) = 25
* Sample proportion (p̂) = x/n = 25/311 ≈ 0.0804

**4. Calculate the Test Statistic**

We'll use the z-test for proportions:

$$ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} $$

Where:

* p̂ = sample proportion (0.0804)
* p₀ = hypothesized proportion (0.10)
* n = sample size (311)

$$ z = \frac{0.0804 - 0.10}{\sqrt{\frac{0.10(1-0.10)}{311}}} $$

$$ z = \frac{-0.0196}{\sqrt{\frac{0.09}{311}}} $$

$$ z = \frac{-0.0196}{\sqrt{0.000289389}} $$

$$ z = \frac{-0.0196}{0.01701144} $$

$$ z \approx -1.1522 $$

**5. Calculate the P-value**

Since this is a left-tailed test, we need to find the area to the left of z = -1.1522 in the standard normal distribution.

Using a z-table or calculator, we find:

* P(Z < -1.1522) ≈ 0.1246

Therefore, the p-value is approximately 0.1246.

**6. Make a Decision**

* Compare the p-value (0.1246) with the significance level (0.01).
* Since 0.1246 > 0.01, we fail to reject the null hypothesis.

**7. Conclusion**

There is not sufficient evidence at the 0.01 significance level to support the claim that less than 10 percent of the test results are wrong.

**Summary**

* **Hypothesis test:** Left-tailed z-test for proportions.
* **p-value:** approximately 0.1246.