Cassandra's Cogs advertises on its website that 90% of customer orders are received within three working days. They performed an audit from a random sample of 100 of the 2,500 orders that month and it shows 85 orders were received on time.

Part A: Can we use a normal approximation? Explain. (5 points)

Part B: If Cassandra's Cogs customers receive 90% of their orders within three working days, what is the probability that the proportion in the random sample of 100 orders is the same as the proportion found in the audit sample or less? (5 points) (10 points)

Using the normal distribution and the central limit theorem, it is found that:

In a normal distribution with mean and standard deviation , the z-score of a measure X is given by:

In this problem:

The mean and the standard error are given by:

Item a:

Since both and , a normal approximation can be used.

Item b:

The probability is the p-value of Z when , hence:

By the Central Limit Theorem

has a p-value of 0.0475.

There is a 0.0475 = 4.75% probability that the proportion in the random sample of 100 orders is the same as the proportion found in the audit sample or less.

To learn more about the normal distribution and the central limit theorem, you can take a look at link

Part A: yes

Part B: 50%

Step-by-step explanation:

There are two ways we can use a normal approximation for a sample: if the population it is from is normally distributed, or if the sample is sufficiently large (n > 30).  In this case, we don't know if the population is normally distributed, but we do know the sample size is larger than 30, so we can use normal approximation.

The sample proportion is normally distributed, with a mean equal to the mean of the population.  So there is a 50% chance that sample proportion is less than the population proportion.

a.

step-by-step explanation:

step 1 distribute 2 to every term in the parentheses

step 2 add 10 on both sides of the equal sign