Question

Suppose 35.45% of small businesses experience cash flow problems in their first 5 years. A consultant takes a random sample of 530 businesses that have been opened for 5 years or less. What is the probability that between 34.2% and 39.03% of the businesses have experienced cash flow problems?

1) 0.6838
2) 20.3738
3) 0.3162
4) - 11.6695
5) 1.2313

Answer 1

1) 0.6838

Explanation:

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

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes 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 = (\sigma)/(√(n))`.

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

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

35.45% of small businesses experience cash flow problems in their first 5 years.

This means that
`p = 0.3545`

Sample of 530 businesses

This means that
`n = 530`

Mean and standard deviation:

`\mu = p = 0.3545`

`s = \sqrt{(p(1-p))/(n)} = \sqrt{(0.3545(1-0.3545))/(530)} = 0.0208`

What is the probability that between 34.2% and 39.03% of the businesses have experienced cash flow problems?

This is the p-value of Z when X = 0.3903 subtracted by the p-value of Z when X = 0.342.

X = 0.3903

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

By the Central Limit Theorem

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

`Z = (0.3903 - 0.3545)/(0.0208)`

`Z = 1.72`

`Z = 1.72` has a p-value of 0.9573

X = 0.342

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

`Z = (0.342 - 0.3545)/(0.0208)`

`Z = -0.6`

`Z = -0.6` has a p-value of 0.27425

0.9573 - 0.2743 = 0.683

With a little bit of rounding, 0.6838, so option 1) is the answer.