An instructor gives a 100-point examination in which the...

An instructor gives a 100-point examination in which the grades are normally distributed. The mean is 60 and the standard deviation is $10 .$ If there are $5 \%$ A's and $5 \% \mathrm{~F}^{\prime} \mathrm{s}, 15 \% \mathrm{~B}^{\prime} \mathrm{s}$ and $15 \% \mathrm{D}^{\prime} \mathrm{s},$ and $60 \% \mathrm{C}^{\prime} \mathrm{s},$ find the scores that divide the distribution into those categories.

Key Concepts

Inverse Cumulative Distribution Function

The inverse cumulative distribution function, also known as the quantile function, is used to determine the value (or cutoff) corresponding to a given cumulative probability in a probability distribution. In the context of the normal distribution, it is applied to find the required z-scores that then, through back-transformation, yield the specific cutoff values for desired percentiles.

Standardization

Standardization is the process of converting data from a normal distribution to a standard normal distribution by subtracting the mean and dividing by the standard deviation. This transformation simplifies analysis by allowing the use of the standard normal table, which provides cumulative probabilities associated with z-scores.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about the mean. It is defined by two parameters: the mean (which indicates the center of the distribution) and the standard deviation (which measures the spread of the data). Many real-world phenomena approximate this distribution, making it fundamental in statistics for modeling and inference.

Percentiles

Percentiles are values in a data distribution that divide the dataset into 100 equal parts. They represent the relative standing of a value within a distribution. In the context of the normal distribution, percentiles are used to determine cutoff points that categorize data into groups, such as grades or performance levels, based on cumulative probabilities.

Z-Scores

A z-score is a standardized score that indicates how many standard deviations a data point is from the mean of the distribution. It allows for comparison between different data points and is used to transform a normal distribution into a standard normal distribution (with a mean of 0 and a standard deviation of 1), facilitating the use of standard tables or statistical software to find probabilities.

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