Question

Based on historical data, your manager believes that 34% of the company's orders come from first-time customers. A random sample of 137 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is less than 0.27?

Note: You should carefully round any z-values you calculate to 4 decimal places to match wamap's approach and calculations.

Answer =
(Enter your answer as a number accurate to 4 decimal places.)

Answer 1

To determine the probability that the sample proportion is less than 0.27, we can use the normal approximation of the binomial distribution.

1. **Define the known parameters**:
- Population proportion (\( p \)): 0.34
- Sample size (\( n \)): 137
- Sample proportion to evaluate (\( \hat{p} \)): 0.27

2. **Calculate the mean and standard deviation of the sampling distribution**:
- The mean of the sampling distribution \( \mu_{\hat{p}} \) is given by:
\[
\mu_{\hat{p}} = p = 0.34
\]
- The standard deviation of the sampling distribution \( \sigma_{\hat{p}} \) is given by:
\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.34(1-0.34)}{137}}
\]

First, calculate \( 1 - p \):
\[
1 - p = 1 - 0.34 = 0.66
\]

Now compute \( \sigma_{\hat{p}} \):
\[
\sigma_{\hat{p}} = \sqrt{\frac{0.34 \times 0.66}{137}} = \sqrt{\frac{0.2244}{137}} \approx \sqrt{0.00164} \approx 0.0405
\]

3. **Calculate the z-score**:
The z-score for the sample proportion \( \hat{p} = 0.27 \) is calculated as:
\[
z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.27 - 0.34}{0.0405}
\]
\[
z = \frac{-0.07}{0.0405} \approx -1.7284
\]

4. **Find the probability**:
We will look up the z-score in the standard normal distribution table (or use a calculator). The z-score of -1.7284 corresponds to a probability.

Using standard normal distribution tables or calculators gives us:
\[
P(Z < -1.7284) \approx 0.04211
\]

So, the probability that the sample proportion is less than 0.27 is approximately:

\[
\text{Answer} = 0.0421
\]

Thus, your answer is:

\[
\boxed{0.0421}
\]