Question

Dartmouth Distribution Warehouse makes deliveries of a large number of products to its customers. It is known that 78% of all the orders it receives from its customers are delivered on time. Let o be the proportion of orders in a random sample of 100 that are delivered on time. Find the probability that the value of P will be less than 0.81

Answer 1

Answer:

The probability that the value of P will be less than 0.81 is 0.7642.

Step-by-step explanation:

According to the Central limit theorem, if from an unknown population large samples of sizes n > 30, are selected and the sample proportion for each sample is computed then the sampling distribution of sample proportion follows a Normal distribution.

The mean of this sampling distribution of sample proportion is:

 $\mu_{\hat p}=p$

The standard deviation of this sampling distribution of sample proportion is:

$\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n}}$

Given:

n = 100

p = 0.78

Since n = 100 > 30, according to the central limit theorem the sampling distribution of sample proportion follows a Normal distribution.

$\hat p\sim N(\mu_{\hat p}=0.78, \sigma_{\hat p}=0.0414)$

Compute the value of $P(\hat p$ as follows:

$P(\hat p$

                  $=P(Z$

*Use a z-table for the probability.

Thus, the probability that the value of P will be less than 0.81 is 0.7642.