The problem asks us to determine the probability that the sample proportion of first-time customers is greater than 0.3. Here's how we can approach this step-by-step:

Given information:

• The population proportion $p = 0.25$ (25% of orders come from first-time customers).
• Sample size $n = 73$.
• The sample proportion $\hat{p} = 0.3$.
• We are asked to find the probability that the sample proportion is greater than 0.3, i.e., $P(\hat{p} > 0.3)$.

Solution Approach:

We will use the normal approximation to the binomial distribution for large sample sizes. According to the Central Limit Theorem, for large $n$, the sampling distribution of the sample proportion $\hat{p}$ will be approximately normally distributed with the following properties:

• Mean of the sampling distribution: $\mu_{\hat{p}} = p = 0.25$
• Standard deviation (standard error) of the sampling distribution: $\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.25(1 - 0.25)}{73}} = \sqrt{\frac{0.25 \times 0.75}{73}} \approx 0.0504$

Now, we standardize the sample proportion $\hat{p} = 0.3$ to convert it to a standard normal variable $Z$.

• Z-score formula: $Z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.3 - 0.25}{0.0504} \approx 0.9921$

Using the Z-score of 0.9921, we now find the probability from standard normal distribution tables (or using a calculator).

• P-value for Z = 0.9921 is approximately 0.8397 (from Z-tables or using a normal distribution calculator).

Thus, the probability that $\hat{p}$ is greater than 0.3 is: $P(\hat{p} > 0.3) = 1 - P(Z \leq 0.9921) = 1 - 0.8397 = 0.1603$

Final Answer:

The probability that the sample proportion is greater than 0.3 is approximately 0.1603.

Would you like further details on any step? Here are some related questions for further exploration:

1. How would the probability change if the sample size was larger?
2. What is the effect of increasing the population proportion $p$?
3. Why do we use the normal approximation in this problem?
4. How do we calculate the Z-score for different sample proportions?
5. What is the interpretation of a Z-score?

Tip:

Always check if the sample size $n$ is large enough (typically $np \geq 5$ and $n(1-p) \geq 5$) before using the normal approximation to the binomial distribution.