Mathematics, 08.04.2020 02:16 vnfrancis3716

Professor Elderman has given the same multiple-choice final exam in his Principles of Microeconomics class for many years. After examining his records from the past 10 years, he finds that the scores have a mean of 76 and a standard deviation of 12. What is the probability that a class of 15 students will have a class average greater than 70 on Professor Elderman's final exam?

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97.38% probability that a class of 15 students will have a class average greater than 70 on Professor Elderman's final exam

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.

In this problem, we have that:

$\mu = 76, \sigma = 12, n = 15, s = \frac{12}{\sqrt{15}} = 3.0984$

What is the probability that a class of 15 students will have a class average greater than 70 on Professor Elderman's final exam?

This is 1 subtracted by the pvalue of Z when X = 70. So

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

By the Central Limit Theorem

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

$Z = \frac{70 - 76}{3.0984}$

Z = -1.94

Z = -1.94 has a pvalue of 0.0262

1 - 0.0262 = 0.9738

97.38% probability that a class of 15 students will have a class average greater than 70 on Professor Elderman's final exam

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Professor Elderman has given the same multiple-choice final exam in his Principles of Microeconomics...

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