This problem involves calculating the probability that the sample proportion of first-time customers is less than 0.3, given that 25% (or 0.25) of the company's orders come from first-time customers. We will solve this using the normal approximation to the binomial distribution.

Steps:

1. Identify the given parameters:

   • $p = 0.25$ (the population proportion of first-time customers)
   • $n = 54$ (the sample size)
   • We are interested in finding the probability that the sample proportion $\hat{p}$ is less than 0.3.

2. Find the mean and standard deviation of the sampling distribution of $\hat{p}$:

   • The mean of the sampling distribution of $\hat{p}$ is: $\mu_{\hat{p}} = p = 0.25$
   • The standard deviation of the sampling distribution of $\hat{p}$ is: $\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.25(1 - 0.25)}{54}} = \sqrt{\frac{0.25 \times 0.75}{54}} = \sqrt{\frac{0.1875}{54}} \approx 0.0587$

3. Convert the sample proportion to a z-score:

   • We need to convert $\hat{p} = 0.3$ to a z-score using the formula: $z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.3 - 0.25}{0.0587} \approx \frac{0.05}{0.0587} \approx 0.8517$

4. Find the probability corresponding to the z-score:

   • Using a standard normal table or calculator, we find the probability that $z$ is less than 0.8517. From the table, the cumulative probability for $z = 0.8517$ is approximately 0.8023.

Thus, the probability that the sample proportion is less than 0.3 is approximately 0.8023.

Answer:

$\boxed{0.8023}$

Would you like further clarification or additional details?

Related Questions:

1. What if the population proportion was 0.3 instead of 0.25?
2. How does increasing the sample size affect the probability calculation?
3. What if we wanted the probability that the sample proportion is greater than 0.3?
4. How does the standard deviation of the sampling distribution change with different sample sizes?
5. What is the probability that the sample proportion is exactly 0.3?

Tip:

When using the normal approximation to the binomial distribution, ensure the sample size is large enough, typically when $np \geq 5$ and $n(1 - p) \geq 5$.