10-5 The normal distribution

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Presentation transcript:

10-5 The normal distribution

objective I can use the Empirical Rule to analyze normally distributed variables. I can apply the standard normal distribution and z-values.

Normal distribution curve A normal distribution curve is a continuous, symmetric, bell-shaped distribution of a random variable. It is the most common continuous probability distribution.

The normal distribution The graph of the curve is continuous, bell-shaped, and symmetric with respect to the mean. The mean, median, and mode are equal and located at the center. The total area under the curve is equal to 1 or 100%. The area under the normal curve represents the amount of data within a certain interval or the probability that a random data value falls within that interval. The Empirical Rule can be used to determine the area under the normal curve at specific intervals.

The empirical rule

A normal distribution has a mean of 21 and a standard deviation of 4. Example 1 A normal distribution has a mean of 21 and a standard deviation of 4. Find the range of values that represent the middle 68% of the distribution. What percent of the data will be greater than 29?

Example 2 A normal distribution has a mean of 8.2 and a standard deviation of 1.3. Find the range of values that represent the middle 95% of the distribution. What percent of the data will be less than 4.3?

Example 3 The heights of 1800 adults are normally distributed with a mean of 70 inches and a standard deviation of 2 inches. About how many adults are between 66 and 74 inches? What is the probability that a random adult is more than 72 inches tall?

Standard normal distribution

Example 4 Find z if X = 18, 𝜇=22, and 𝜎=3.1. Indicate the position of X in the distribution.

Example 5 The number of videos uploaded daily to a video-sharing site is normally distributed with 𝜇=181,099 videos and 𝜎=35,644 videos. Find each probability. Use a z-table to calculate the area under the curve. P(180,000 < X < 200,000) P(X > 250,000)