To calculate the test statistic $z$ for this hypothesis test, we will use the formula for the one-sample z-test for proportions:

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

Where:

• $\hat{p}$ is the sample proportion (the proportion of satisfied customers in the survey),
• $p_0$ is the hypothesized population proportion (the current claimed satisfaction rate, which is 76% or 0.76),
• $n$ is the sample size (the number of surveyed customers).

Step 1: Calculate the sample proportion $\hat{p}$

From the survey:

• 170 customers out of 215 were satisfied.

$\hat{p} = \frac{170}{215} = 0.7907$

Step 2: Calculate the standard error

The standard error is given by:

$\text{Standard Error} = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.76(1 - 0.76)}{215}} = \sqrt{\frac{0.76 \times 0.24}{215}} = \sqrt{\frac{0.1824}{215}} = \sqrt{0.000849} \approx 0.02914$

Step 3: Compute the z-value

Now, substitute the values into the z-test formula:

$z = \frac{0.7907 - 0.76}{0.02914} = \frac{0.0307}{0.02914} \approx 1.05$

Final Answer:

The test statistic $z$ is approximately 1.05 (rounded to two decimal places).

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Would you like more details on the hypothesis testing process or further steps in interpreting the result?

Related Questions:

1. How do we interpret the z-value in terms of a p-value?
2. What is the meaning of the claimed 76% satisfaction rate in hypothesis testing?
3. How do you conduct a hypothesis test for population proportions?
4. What is the concept of Type I and Type II errors in hypothesis testing?
5. How would the z-test change if the sample size were different?

Tip: Always double-check the conditions for using the z-test, like having a sufficiently large sample size and the number of successes and failures being both greater than 5.