The distribution of the scores on a certain exam is N(80,5)...

The distribution of the scores on a certain exam is $N(80,5)$ which means that the exam scores are Normally distributed with a mean of 80 and a standard deviation of 5 . a. Sketch or use technology to create the curve and label on the $x$ -axis the position of the mean, the mean plus or minus one standard deviation, the mean plus or minus two standard deviations, and the mean plus or minus three standard deviations. b. Find the probability that a randomly selected score will be greater than 90. Shade the region under the Normal curve whose area corresponds to this probability.

The distribution of the scores on a certain exam is $N(80,5)$ which means that the exam scores are Normally distributed with a mean of 80 and a standard deviation of 5 .
a. Sketch or use technology to create the curve and label on the $x$ -axis the position of the mean, the mean plus or minus one standard deviation, the mean plus or minus two standard deviations, and the mean plus or minus three standard deviations.
b. Find the probability that a randomly selected score will be greater than
90. Shade the region under the Normal curve whose area corresponds to this probability.

Key Concepts

Normal Distribution

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 often depicted as a bell-shaped curve and is defined fully by its mean and standard deviation.

Mean

The central value of a distribution, found by averaging all the data points. In a normal distribution, the mean is the point around which the curve is centered and represents the expected value of the data.

Standard Deviation

A measure of the dispersion or spread in a set of values that quantifies how much the values typically deviate from the mean. It determines the width of the bell curve in a normal distribution.

Z-score

A statistical measurement that describes a value's position relative to the mean of a group of values, measured in terms of standard deviations. It is used to calculate probabilities in a normal distribution by standardizing different distributions to a common scale.

Area Under the Curve

The integral of the probability density function over an interval, representing the probability that a randomly selected observation lies within that interval. In a normal distribution, the total area under the curve is equal to one.

Sketching the Normal Curve

The process of drawing the bell-shaped curve of a normal distribution, including annotating the key points such as the mean, and intervals of one, two, and three standard deviations from the mean to provide a visual understanding of the spread and distribution of the data.

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