Scores on the common final exam given in a large enrollment...

Scores on the common final exam given in a large enrollment multiple section course were normally distributed with mean 69.35 and standard deviation $12.93 .$ The department has the rule that in order to receive an A in the course his score must be in the top $10 \%$ of all exam scores. Find the minimum exam score that meets this requirement.

Scores on the common final exam given in a large enrollment multiple section course were normally
distributed with mean 69.35 and standard deviation $12.93 .$ The department has the rule that in order to
receive an A in the course his score must be in the top $10 \%$ of all exam scores. Find the minimum exam score
that meets this requirement.

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, mean, and standard deviation, and it is widely used because many phenomena naturally follow this pattern.

Percentiles

Percentiles divide a data set into 100 equal parts, indicating the relative standing of a value within the distribution. In the context of the normal distribution, percentiles help determine the threshold values below which a specified percentage of data falls, making them essential for setting cutoff scores or criteria.

Standardization and Z-scores

Standardization converts raw data values from any normal distribution into z-scores, which represent the number of standard deviations a data point is from the mean. This process allows different normal distributions to be compared on a common scale, typically the standard normal distribution with a mean of 0 and a standard deviation of 1.

Inverse Normal Calculation

The inverse normal calculation involves using the cumulative distribution function in reverse to find the raw score corresponding to a specific percentile. This is done by first finding the z-score associated with the desired cumulative probability and then converting that z-score back to the original scale using the distribution’s mean and standard deviation.

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