Mathematics, 23.03.2020 21:33 mothertrucker2828

For a standardized psychology examination intended for psychology majors, the historical data show that scores have a mean of 500 and a standard deviation of 180. The grading process of this year's exam has just begun. The average score of the 35 exams graded so far is 528. What is the probability that a sample of 35 exams will have a mean score of 528 or more if the exam scores follow the same distribution as in the past?

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17.88% probability that a sample of 35 exams will have a mean score of 528 or more if the exam scores follow the same distribution as in the past

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:

$\mu = 500, \sigma = 180,n = 35, s = \frac{180}{\sqrt{35}} = 30.4256$

What is the probability that a sample of 35 exams will have a mean score of 528 or more if the exam scores follow the same distribution as in the past?

This is 1 subtracted by the pvalue of Z when X = 528. So

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

By the Central Limit Theorem

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