Question

Business Weekly conducted a survey of recent graduates from the top MBA programs. On the basis of the survey, assume that 60% of the recent graduates annual salary exceeds $40000. Suppose you take a simple random sample of 51 recent graduates. Note: round all z scores to decimal places and all other numbers to 4 decimal places. Find the probability that less than 70% of the 51 recent graduates have an annual salary exceeding $40000

Answer 1

Using the normal distribution and the central limit theorem, it is found that there is a 0.9279 = 92.79% probability that less than 70% of the 51 recent graduates have an annual salary exceeding $40000.

Normal Probability Distribution

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

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

• It 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, which is the percentile of X.

• By the Central Limit Theorem, for a proportion p in a sample of size n, the sampling distribution of sample proportion is approximately normal with mean $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1 - p)}{n}}$, as long as $np \geq 10$ and $n(1 - p) \geq 10$.

In this problem, we have that:

• 60% of the recent graduates annual salary exceeds $40000, hence p = 0.6.

• A sample of 51 is taken, hence n = 51.

The mean and the standard error are given by:

$\mu = p = 0.6$

$s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.6(0.4)}{51}} = 0.0686$

The probability that less than 70% of the 51 recent graduates have an annual salary exceeding $40000 is the p-value of Z when X = 0.7, hence:

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

By the Central Limit Theorem

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

$Z = \frac{0.7 - 0.6}{0.0686}$

$Z = 1.46$

$Z = 1.46$ has a p-value of 0.9279.

0.9279 = 92.79% probability that less than 70% of the 51 recent graduates have an annual salary exceeding $40000.

To learn more about the normal distribution and the central limit theorem, you can check https://brainly.com/question/24663213