Achievement test scores from all high-school seniors in a certain state have a mean and variance of 60 and 64, respectively. A specific high school class of n=100 students had a mean score of 58. Is there evidence to suggest that this high school is substandard? (Consider calculating the probability that the sample mean is at most 58 when n=100)

0.62% that the sample mean is at most 58. Any probability lower than 5% is considered unusually low, which means that there is evidence to suggest that this high school is substandard.

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 random variable X, with mean and standard deviation , a large sample size can be approximated to a normal distribution with mean and standard deviation

In this problem, we have that:

Also important to remember that the standard deviation is the square root of the variance. So

A specific high school class of n=100 students had a mean score of 58. Is there evidence to suggest that this high school is substandard? (Consider calculating the probability that the sample mean is at most 58 when n=100)

This probability is the pvalue of Z when X = 58. So

By the Central Limit Theorem

has a pvalue of 0.0062.

So there is a 0.62% that the sample mean is at most 58. Any probability lower than 5% is considered unusually low, which means that there is evidence to suggest that this high school is substandard.

109 is the nearest whole number/final answer! : )