To determine whether there is sufficient evidence at the 0.05 level to dispute the company's claim, we conduct a hypothesis test for a population proportion. Here's a step-by-step explanation:

Step 1: Set up the hypotheses

• Null Hypothesis ($H_0$): $p = 0.39$ (The population proportion of failed chips is 39%.)
• Alternative Hypothesis ($H_a$): $p \neq 0.39$ (The population proportion of failed chips is not 39%.)

This is a two-tailed test since we are looking for a difference in either direction.

Step 2: Identify the sample statistics

• Sample proportion ($\hat{p}$) = $36\%$ or $0.36$
• Sample size ($n$) = $1100$
• Claimed proportion ($p_0$) = $0.39$

Step 3: Calculate the test statistic

The test statistic for a proportion is given by:

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

Substituting the values:

$z = \frac{0.36 - 0.39}{\sqrt{\frac{0.39(1 - 0.39)}{1100}}}$

First, calculate the denominator:

$\sqrt{\frac{0.39(1 - 0.39)}{1100}} = \sqrt{\frac{0.39 \times 0.61}{1100}} = \sqrt{\frac{0.2379}{1100}} = \sqrt{0.00021627} \approx 0.0147$

Now calculate the test statistic:

$z = \frac{0.36 - 0.39}{0.0147} = \frac{-0.03}{0.0147} \approx -2.04$

Step 4: Determine the critical value

For a two-tailed test at the $0.05$ significance level, the critical z-values are:

$z = \pm 1.96$

Step 5: Make a decision

• If $z$-statistic is less than $-1.96$ or greater than $1.96$, we reject $H_0$.
• Here, $z \approx -2.04$, which is less than $-1.96$.

Step 6: Conclusion

Since the test statistic falls in the rejection region, we reject the null hypothesis. There is sufficient evidence at the $0.05$ level to dispute the company's claim that 39% of the chips fail in the first 1000 hours.

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Do you want additional details or explanations about this process?

Related Questions:

1. How is the formula for the test statistic derived?
2. What would happen if the significance level were changed to $0.01$?
3. How does the sample size affect the result of this hypothesis test?
4. Can we calculate the p-value for this test? If yes, how?
5. How would a one-tailed test change the outcome of this problem?

Tip:

When conducting hypothesis tests, always visualize the critical regions of the z-distribution to better understand the rejection and non-rejection zones