Questions: Formulas: np ; sqrt(np(1-p)) ; sqrt(p(1-p)/n) ; sigma/sqrt(n) Based on historical data, your manager believes that 34% of the company's orders come from first-time customers. A random sample of 90 orders will be used to estimate the proportion of first-time-customers. Show your answer to 4 decimal places if necessary. a. What is the distribution of p-hat ? p-hat-N() b. What is the probability that the sample proportion is more than 0.41 ? c. What is the probability that the sample proportion is between 0.35 and 0.39 ?

Formulas: np ; sqrt(np(1-p)) ; sqrt(p(1-p)/n) ; sigma/sqrt(n)
Based on historical data, your manager believes that 34% of the company's orders come from first-time customers. A random sample of 90 orders will be used to estimate the proportion of first-time-customers. Show your answer to 4 decimal places if necessary.
a. What is the distribution of p-hat ? p-hat-N()
b. What is the probability that the sample proportion is more than 0.41 ?
c. What is the probability that the sample proportion is between 0.35 and 0.39 ?

Transcript text: Formulas: $n p ; \quad \sqrt{n p(1-p)} ; \sqrt{\frac{p(1-p)}{n}} ; \frac{\sigma}{\sqrt{n}}$ Based on historical data, your manager believes that $34 \%$ of the company's orders come from first-time customers. A random sample of 90 orders will be used to estimate the proportion of first-time-customers. Show your answer to 4 decimal places if necessary. a. What is the distribution of $\hat{p} ? \hat{p}-\mathrm{N}($ $\square$ $\square$ ) b. What is the probability that the sample proportion is more than 0.41 ? $\square$ c. What is the probability that the sample proportion is between 0.35 and 0.39 ? $\square$

Solution

Solution Steps

Solution Approach

a. The distribution of the sample proportion $\hat{p}$ is approximately normal with mean equal to the population proportion $p$ and standard deviation $\sqrt{\frac{p(1-p)}{n}}$, where $n$ is the sample size.

b. To find the probability that the sample proportion is more than 0.41, calculate the z-score for 0.41 using the mean and standard deviation from part (a), and then find the corresponding probability using the standard normal distribution.

c. To find the probability that the sample proportion is between 0.35 and 0.39, calculate the z-scores for both 0.35 and 0.39, and then find the probability between these z-scores using the standard normal distribution.

Step 1: Determine the Distribution of $\hat{p}$

The sample proportion $\hat{p}$ follows a normal distribution with mean $\mu_{\hat{p}} = p$ and standard deviation $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$.

Given:

$p = 0.34$
$n = 90$

Calculate the standard deviation: \[ \sigma_{\hat{p}} = \sqrt{\frac{0.34 \times (1 - 0.34)}{90}} \approx 0.04993 \]

Thus, the distribution of $\hat{p}$ is $\hat{p} \sim N(0.34, 0.04993)$.

Step 2: Probability that $\hat{p} > 0.41$

Calculate the z-score for $\hat{p} = 0.41$: \[ z = \frac{0.41 - 0.34}{0.04993} \approx 1.4019 \]

Find the probability: \[ P(\hat{p} > 0.41) = 1 - P(Z \leq 1.4019) \approx 0.08048 \]

Step 3: Probability that $0.35 < \hat{p} < 0.39$

Calculate the z-scores for $\hat{p} = 0.35$ and $\hat{p} = 0.39$: \[ z_{0.35} = \frac{0.35 - 0.34}{0.04993} \approx 0.2003 \] \[ z_{0.39} = \frac{0.39 - 0.34}{0.04993} \approx 1.0013 \]

Find the probability: \[ P(0.35 < \hat{p} < 0.39) = P(Z \leq 1.0013) - P(Z \leq 0.2003) \approx 0.2623 \]