Question

The head of the Westlane Cultural Center wants to get a sense of how quickly pledges from donors arrive at the center. It takes an average of 28 days, with a standard deviation of 7 days (normal distribution), from the time pledges are made until donations are actually received. To better manage cash flow, the director wants to know what the status of pledges looks like by about 40 days. What is the probability that pledges are received within 40 days

Answer 1

95.64% probability that pledges are received within 40 days

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 28, \sigma = 7`

What is the probability that pledges are received within 40 days

This is the pvalue of Z when X = 40. So

`Z = (X - \mu)/(\sigma)`

`Z = (40 - 28)/(7)`

`Z = 1.71`

`Z = 1.71` has a pvalue of 0.9564

95.64% probability that pledges are received within 40 days