To find the p-value for this hypothesis test, we need to follow the steps for a one-sample z-test for proportions. Let's break it down:

Step 1: Define the problem

You are testing whether the proportion of online orders is greater than 75% (0.75). The null and alternative hypotheses are:

• $H_0: p \leq 0.75$ (the proportion of online orders is less than or equal to 75%)
• $H_a: p > 0.75$ (the proportion of online orders is greater than 75%)

Step 2: Collect the sample data

From the sample of 1,000 orders, 745 were online orders. So:

• $\hat{p} = \frac{745}{1000} = 0.745$
• $n = 1000$

Step 3: Calculate the test statistic

We use the formula for the z-test statistic for proportions: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$ where:

• $\hat{p} = 0.745$ (sample proportion)
• $p_0 = 0.75$ (hypothesized population proportion)
• $n = 1000$ (sample size)

Substitute the values into the formula: $z = \frac{0.745 - 0.75}{\sqrt{\frac{0.75(1 - 0.75)}{1000}}}$ $z = \frac{-0.005}{\sqrt{\frac{0.75 \times 0.25}{1000}}}$ $z = \frac{-0.005}{\sqrt{\frac{0.1875}{1000}}}$ $z = \frac{-0.005}{\sqrt{0.0001875}} \approx \frac{-0.005}{0.0137} \approx -0.365$

Step 4: Find the p-value

Now that we have the z-value, we look up the p-value for $z = -0.365$ in the standard normal table.

Since the alternative hypothesis is $p > 0.75$, this is a right-tailed test. The p-value corresponds to the area under the curve to the right of $z = -0.365$.

The area corresponding to $z = -0.365$ from the standard normal table is approximately 0.643. Therefore, the area to the right of $z = -0.365$ is: $p\text{-value} = 1 - 0.643 = 0.357$

Final Answer:

The p-value for this hypothesis test is approximately 0.357 (rounded to three decimal places).

Let me know if you would like further clarification or have any questions!

Here are 5 related questions for you to explore:

1. What would happen to the p-value if the sample size were increased?
2. How do you interpret a p-value of 0.357 in the context of hypothesis testing?
3. What changes would you make if the alternative hypothesis were two-tailed?
4. How would you calculate the margin of error for the proportion of online orders?
5. Why is the z-test appropriate for this hypothesis test, and when should you use other tests?

Tip: A p-value greater than the significance level (usually 0.05) means that we do not reject the null hypothesis, suggesting insufficient evidence to support the claim that the proportion of online orders is greater than 75%.