Normal Distributions

The people in this photograph are a sample of a population and a source of valuable data. Like a lot of data on natural phenomena, people’s heights and weights fit a “normal distribution.” Medical statisticians use this information to plot height and weight charts and to establish guidelines on healthy weight.

Normal Distributions The information can also be used to chart changes in a population over time. For example, the data can be analyzed to determine whether people, on the whole, are getting taller or heavier. These results may affect or even determine government health policy. Moreover, manufacturing and other industries may use the information to decide whether to, for example, make door frames taller or airplane seats wider.

Normal Distributions The information can also be used to chart changes in a population over time. For example, the data can be analyzed to determine whether people, on the whole, are getting taller or heavier. These results may affect or even determine government health policy. Moreover, manufacturing and other industries may use the information to decide whether to, for example, make door frames taller or airplane seats wider.

Normal Distributions The normal distribution is the most important distribution for a continuous random variable. Many naturally occurring phenomena have a distribution that is normal or approximately normal. Some examples are: Physical attributes of a population, such as height, weight, and arm length Crop yields Scores for tests taken by a large population Once a normal model has been established, we can use it to make predictions about a distribution and to answer other relevant questions.

How a Normal Distribution Arises Consider oranges picked from an orange tree. They do not all have the same weight. The variation may be due to several factors, including: Genetics Different times when the flowers were fertilized Different amounts of sunlight reaching the leaves and fruit Different weather conditions such as heavy winds The result is that most of the oranges will have weights close to the mean, while fewer oranges will be much heavier or much lighter. This results in a bell-shaped distribution which is symmetric about the mean.

Normal Distributions

Although all normal distributions have the same general bell-shaped curve, the exact location and shape of the curve is determined by the mean, μ, and standard deviation, σ, of the variable. The height of trees in a park is normally distributed with a mean of 10 meters and a standard deviation of 3 meters. The time it takes Sean to get to school is normally distributed with a mean of 15 minutes and a standard deviation of 1 minute.

Normal Distributions

The x-axis is an asymptote to the curve. The total area under the curve is 1 (or 100%). 50% of the data is to the left of the mean, and 50% of the data is to the right of the mean.

Empirical Rule

Your textbook uses 68.26%, 95.44%, and 99.74% respectively for the empirical rule. You can use those values, or you can just use 68%, 95%, and 99.7%. Depending on the values you use, you may not get exactly the answer in the back of the book.

Normal Distributions You can use normal distributions to determine an expected value. Expected value: found by multiplying the number in the sample by the probability.

Normal Distributions Example 1. The waiting times for an elevator are normally distributed with a mean of 1.5 minutes and a standard deviation of 20 seconds. a. Sketch a normal distribution diagram to illustrate this information, clearly indicating the mean, and the times within 1, 2, and 3 standard deviations of the mean.

Normal Distributions Example 1. The waiting times for an elevator are normally distributed with a mean of 1.5 minutes and a standard deviation of 20 seconds. b. Find the probability that a person waits longer than 2 minutes 10 seconds for the elevator. P(waiting longer than 2 minutes 10 seconds) = 2.5% or 0.25

Normal Distributions Example 1. The waiting times for an elevator are normally distributed with a mean of 1.5 minutes and a standard deviation of 20 seconds. c. Find the probability that a person waits less than 1 minute 10 seconds for the elevator. P(waiting less than 1 minutes 10 seconds) = 16% or 0.16

Normal Distributions Example 1. The waiting times for an elevator are normally distributed with a mean of 1.5 minutes and a standard deviation of 20 seconds. d. 200 people are observed and the length of time they wait for an elevator is noted. Calculate the number of people expected to wait less than 50 seconds for the elevator.

Normal Distributions Example 1. The waiting times for an elevator are normally distributed with a mean of 1.5 minutes and a standard deviation of 20 seconds. d. 200 people are observed and the length of time they wait for an elevator is noted. Calculate the number of people expected to wait less than 50 seconds for the elevator.

Normal Distributions Example 2. The heights of 250 twenty-year-old women are normally distributed with a mean of 1.68m and a standard deviation of 0.06m. a. Sketch a normal distribution diagram to illustrate this information, clearly indicating the mean, and the heights within 1, 2, and 3 standard deviations of the mean.

Normal Distributions Example 2. The heights of 250 twenty-year-old women are normally distributed with a mean of 1.68m and a standard deviation of 0.06m. b. Find the probability that a woman has a height between 1.56m & 1.74m.

Normal Distributions Example 2. The heights of 250 twenty-year-old women are normally distributed with a mean of 1.68m and a standard deviation of 0.06m. c. Find the expected number of women with a height greater than 1.8m.

Normal Distributions

Technically the lower bound of a normal distribution could be -∞, and the upper bound could be ∞. However, we cannot enter this into our calculator. To enter a very small number, use -1E99. To enter a very large number, use 1E99. To access the “E” button, press “2 nd ” “,”

Normal Distributions

Does it matter if the inequality symbol is vs >? No! We are talking about the probability that an event lies within an interval. When we find the probability, we are finding the area under the curve. The probability that an event exactly equaling a specific value is 0. Think of finding the area of a specific point. We’d multiply length times width, but a specific point has a width of 0, so the area would be 0. P(a < X < b) = P(a < X < b) = P(a < X < b) = P(a < X < b)

Normal Distributions Example 4. The lifetime of a light bulb is normally distributed with a mean of 2,800 hours and a standard distribution of 450 hours. a.Find the percentage of light bulbs that have a lifetime of less than 1,950 hours. normalcdf(-1E99, 1950, 2800, 450) = % of light bulbs have a lifetime of less than 1,950 hours.

Normal Distributions Example 4. The lifetime of a light bulb is normally distributed with a mean of 2,800 hours and a standard distribution of 450 hours. b. Find the percentage of light bulbs that have a lifetime between 2,300 and 3,500 hours. normalcdf(2300, 3500, 2800, 450) = % of light bulbs have a lifetime between 2,300 and 3,500 hours.

Normal Distributions Example 4. The lifetime of a light bulb is normally distributed with a mean of 2,800 hours and a standard distribution of 450 hours. c. Find the percentage of light bulbs that have a lifetime more than 3,800 hours. normalcdf(3800, 1E99, 2800, 450) = % of light bulbs have a lifetime of more than 3,800 hours.

Normal Distributions