If the mean GPA among students is 3.25 with a standard deviation of 1.75, and we select a random sample of 300 people, at what value for the sample mean would be greater than exactly 95% of possible sample means?

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 and standard deviation , the zscore of a measure X is given by:

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 and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

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:

At what value for the sample mean would be greater than exactly 95% of possible sample means?

This is X when Z has a pvalue of 0.95. So X when Z = 1.645.

By the Central Limit Theorem

answ5

step-by-step explanation:

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