Mathematics, 01.07.2021 22:40 ronniethefun

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

Answers: 3

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Answer from: quintinlarrieu

1) 0.6838

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 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:

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

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 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 $s = \frac{\sigma}{\sqrt{n}}$ .

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

For 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 $s = \sqrt{\frac{p(1-p)}{n}}$

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:

$\mu = p = 0.3545$

$s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.3545(1-0.3545)}{530}} = 0.0208$

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

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

By the Central Limit Theorem

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

$Z = \frac{0.3903 - 0.3545}{0.0208}$

Z = 1.72

Z = 1.72 has a p-value of 0.9573

X = 0.342

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

$Z = \frac{0.342 - 0.3545}{0.0208}$

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.

Answer from: Quest

a; x = 0.5

step-by-step explanation:

-10x = -5

x = 0.5

Answer from: Quest

step-by-step explanation:

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