To solve this hypothesis testing problem, let's break it down step by step:

Problem Setup

The company claims the satisfaction rate is $p_0 = 0.76$. A survey is conducted with:

• $n = 268$ (sample size),
• $x = 212$ (number of satisfied customers).

The sample proportion $\hat{p}$ is: $\hat{p} = \frac{x}{n} = \frac{212}{268}$

We are testing:

• Null hypothesis $H_0$: $p = 0.76$
• Alternative hypothesis $H_1$: $p \neq 0.76$ (two-tailed test)

Step 1: Calculate the Test Statistic ($z$)

The formula for the test statistic $z$ is: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$

Step 1.1: Calculate $\hat{p}$

$\hat{p} = \frac{212}{268} \approx 0.791$

Step 1.2: Calculate the standard error

$SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.76 (1 - 0.76)}{268}} = \sqrt{\frac{0.76 \times 0.24}{268}} \approx 0.0267$

Step 1.3: Compute $z$

$z = \frac{\hat{p} - p_0}{SE} = \frac{0.791 - 0.76}{0.0267} \approx 1.16$

So, the test statistic is: $z \approx 1.16$

Step 2: Calculate the p-value

This is a two-tailed test, so we need the cumulative probability of $z = 1.16$ under the standard normal curve. Using a z-table or calculator: $P(Z > 1.16) \approx 0.123$ Since this is two-tailed, multiply by 2: $p\text{-value} = 2 \times 0.123 \approx 0.246$

Step 3: Make a Conclusion

The significance level is $\alpha = 0.01$. Since the p-value ($0.246$) is greater than $0.01$, we fail to reject the null hypothesis. There is insufficient evidence to conclude that the satisfaction rate is different from $76\%$.

Final Answers:

1. Test statistic $z$: $1.16$
2. p-value: $0.246$
3. Conclusion: There is not enough evidence to reject $H_0$ at $\alpha = 0.01$.

Would you like further clarification on any step?
Here are 5 related questions to expand on this topic:

1. How is the p-value interpreted in hypothesis testing?
2. Why do we multiply the tail probability by 2 for a two-tailed test?
3. What happens if the significance level $\alpha$ is changed to $0.05$?
4. How is the sample size $n$ related to the standard error?
5. What are the implications of failing to reject the null hypothesis?

Tip: Always check whether your hypothesis test is one-tailed or two-tailed before calculating the p-value!