Question

Based on historical data, your manager believes that 29% of the company's orders come from first-time customers. A random sample of 169 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is between 0.24 and 0.46

Answer 1

Answer:

Probability that the sample proportion is between 0.24 and 0.46 is 0.936.

Step-by-step explanation:

We are given that based on historical data, your manager believes that 29% of the company's orders come from first-time customers.

A random sample of 169 orders will be used to estimate the proportion of first-time-customers.

Let $\hat p$ = sample proportion

Now, the z score probability distribution for sample proportion is given by;

         Z = $\frac{\hat p -p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }$ ~ N(0,1)

where, p = % of the company's orders that come from first-time customers = 29%

            n = sample of orders = 169

So, probability that the sample proportion is between 0.24 and 0.46 is given by = P(0.24 < $\hat p$ < 0.46) = P($\hat p$ < 0.46) - P($\hat p$ $\leq$ 0.24)

 P($\hat p$ < 0.46) = P( $\frac{\hat p -p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }$ < $\frac{0.46-0.29}{\sqrt{\frac{0.46(1-0.46)}{169} } }$ ) = P(Z < 4.4) = 0.99999

 P($\hat p$ $\leq$ 0.24) = P( $\frac{\hat p -p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }$ $\leq$ $\frac{0.24-0.29}{\sqrt{\frac{0.24(1-0.24)}{169} } }$ ) = P(Z $\leq$ -1.52) = 1 - P(Z < 1.52)    

                                                                = 1 - 0.93574 = 0.06426    

 Therefore, P(0.24 < $\hat p$ < 0.46) = 0.99999 - 0.06426 = 0.936

Hence, probability that the sample proportion is between 0.24 and 0.46 is 0.936.