To determine whether sufficient evidence exists that the customer satisfaction rate is higher than the claimed 50%, we can conduct a hypothesis test for the population proportion. Here's how we go about it step by step:

Step 1: Set up the hypotheses

• Null hypothesis $H_0$: The customer satisfaction rate is equal to the company's claimed rate. This means $p = 0.50$.
• Alternative hypothesis $H_1$: The customer satisfaction rate is higher than the claimed rate. This means $p > 0.50$.

This is a one-tailed test since we are only interested in whether the satisfaction rate is higher than the claimed rate.

Step 2: Compute the sample proportion

The sample proportion $\hat{p}$ is calculated as follows:

$\hat{p} = \frac{\text{number of satisfied customers}}{\text{total number of customers in the sample}} = \frac{130}{244} = 0.5328$

Step 3: Calculate the test statistic

We use the formula for the test statistic for a proportion:

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

Where:

• $\hat{p} = 0.5328$ is the sample proportion,
• $p_0 = 0.50$ is the claimed proportion,
• $n = 244$ is the sample size.

Plugging in the values:

= \frac{0.0328}{\sqrt{\frac{0.25}{244}}} = \frac{0.0328}{0.0319} = 1.28$$ The test statistic is approximately $$z = 1.28$$. ### Step 4: Calculate the p-value Since this is a one-tailed test, the p-value corresponds to the area to the right of $$z = 1.28$$ in the standard normal distribution. Using standard normal distribution tables or a calculator: $$\text{p-value} = P(Z > 1.28) = 0.1003$$ The p-value is approximately $$0.100$$. ### Step 5: Make a decision Let's assume the significance level $$\alpha = 0.05$$. - If the p-value is less than $$\alpha$$, we reject the null hypothesis. - If the p-value is greater than $$\alpha$$, we fail to reject the null hypothesis. Since the p-value of 0.100 is greater than the significance level of 0.05, we **fail to reject the null hypothesis**. This means that there is **not sufficient evidence** to conclude that the customer satisfaction rate is higher than the claimed 50%. ### Conclusion - **Test statistic (z-value):** $$1.28$$ - **p-value:** $$0.100$$ - **Conclusion:** At a 0.05 significance level, there is not enough evidence to suggest that the customer satisfaction rate is higher than 50%. Would you like more details or have any questions? Here are 5 related questions that build on this: 1. What would happen if the sample size was larger? 2. How does changing the significance level affect the conclusion? 3. What is the difference between a one-tailed and two-tailed hypothesis test? 4. How is the p-value interpreted in the context of hypothesis testing? 5. Can you explain what Type I and Type II errors are in hypothesis testing? **Tip:** In hypothesis testing, always check if your test is one-tailed or two-tailed, as it affects how you calculate the p-value.