Question

Forty families gathered for a fund-raising event. Suppose the individual contribution for each family is normally distributed with a mean and a standard deviation of $115 and $35, respectively. The organizers would call this event a success if the total contributions exceed $5,000. What is the probability that this fund-raising event is a success?

Answer 1

Answer:

3.51% probability that this fund-raising event is a success

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

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:

$Z = \frac{X - \mu}{\sigma}$

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.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For the total sum of a sample of size n, we have mean $n*\mu$ and standard deviation $s = \sqrt{n}*\sigma$

In this problem, we have that:

$n = 40, \mu = 40*115 = 4600, \sigma = 35, s = \sqrt{40}*35 = 221.36$

The organizers would call this event a success if the total contributions exceed $5,000. What is the probability that this fund-raising event is a success?

This is 1 subtracted by the pvalue of Z when X = 5000. So

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{5000 - 4600}{221.36}$

$Z = 1.81$

$Z = 1.81$ has a pvalue of 0.9649

1 - 0.9649 = 0.0351

3.51% probability that this fund-raising event is a success