 # The Normal Distribution

## Presentation on theme: "The Normal Distribution"— Presentation transcript:

The Normal Distribution
Density curve Always on or above horizontal axis Area under curve equal to 1 Symmetric density curves have equal mean and median Normal distribution Mean=Median Symmetric, unimodal Area under curve = 1 (100%)

Mean and spread of the normal distribution
Figure 1.28 Introduction to the Practice of Statistics, Sixth Edition © 2009 W.H. Freeman and Company

Density curves with the same mean but different standard deviations.

Empirical Rule(68-95-99.7% Rule)
Approximately 68% of the ordered data will fall within one standard deviation of the mean Approximately 95% of the ordered data will fall within two standard deviations of the mean Approximately 99.7% of the ordered data will fall within three standard deviations of the mean

Empirical Rule: 68-95-99.7% Rule
Figure 1.29 Introduction to the Practice of Statistics, Sixth Edition © 2009 W.H. Freeman and Company

Empirical Rule 34% 34% 13.5% 13.5% 2.35% 2.35% 0.15% 0.15%
Take time to slowly click through slide. Stress that the Empirical Rule can ONLY be used with the assumption that the distribution is normal (bell-shaped curve). Sixty-eight percent of the ordered data of a normal distribution lies within one standard deviation of the mean. Ninety-five percent of the ordered data of a normal distribution lies within two standard deviations of the mean. And, 99.7% of the ordered data of a normal distribution lies within 3 standard deviations of the mean. The normal distribution here, in this example shown, has a mean of 0 and standard deviation of 1. 2.35% 2.35% 0.15% 0.15% How many Standard Deviations away from the mean

1. The average high temperature for the month of April is 65˚F with a standard deviation of 5˚F . Between what values do 68% of April temperatures fall? EXAMPLES: 60 to 70 55 to 75 70 to 80 90 to 100

75 and higher 70 and higher 65 and higher 80 and higher
2. The average high temperature for the month of April is 65˚F with a standard deviation of 5˚F . How high are the highest 2.5% of temperatures for the month of April? 75 and higher 70 and higher 65 and higher 80 and higher

3. The average high temperature for the month of April is 65˚F with a standard deviation of 5˚F % of the temperatures fall into what range? 70 to 80 60 to 70 50 to 80 90 to 100

4. In the summer, a grocery store brings in a large supply of watermelons. The mean weight in pounds is 22. The standard deviation is 4. What percent of watermelons weigh less than 18 pounds? 34% 16% 2.5% 68%

5. In the summer, a grocery store brings in a large supply of watermelons. The mean weight in pounds is 22. The standard deviation is 4. What percent of watermelons weigh more than 30 pounds? 34% 16% 2.5% 68%

6. In the summer, a grocery store brings in a large supply of watermelons. The mean weight in pounds is 22. The standard deviation is 4. What percent of watermelons weigh between 18 and 30 pounds? 34% 81.5% 95% 68%

7. Weights of apples are normally distributed with a mean of 10 oz and a standard deviation of 2 oz.
a.)The middle 68% of apples weigh between _____ and _____. b.)The middle 95% of apples weigh between _____ and _____. c.)The middle 99.7% of apples weigh between _____ and _____. d.) Approximately what percent of apples weigh below 6oz? e.) Approximately what percent of apples weigh above 4 oz?

Z-scores Describes how many standard deviations an observation is from the mean. Negative z-scores (observation is below the mean) Positive z-scores (observation is above the mean) z-score equal to zero (observation is equal to the mean) Standardizes any “score”

Z-scores If we assume the distribution of the variable is normal, then the z-scores have a standard normal distribution.

Examples Find z-score for an apple that weighs 11 oz. 15 oz? 5 oz?

4. The average high temperature for the month of April is 65˚F with a standard deviation of 5˚F Find the standard score of an April high temperature of 71˚F. 1.2 3.5 2.4 5

Standard Normal Distribution
The standard normal distribution has a mean of 0 and a standard deviation of 1. Can use Table A (z-table) to get area under the curve for a standard normal. Area under curve = proportion (percent) Proportions represent probabilities. Examples: (Use the table) What percent of apples weigh below 7 oz? What percent of apples weigh more than 5oz?

Percentiles The cth percentile of a distribution is a value such that c percent of the observations lie below it and the rest lie above.

Example What percentage of April high temperatures fall below 71˚F ?

Example The length of human pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 266 days and standard deviation 16 days. Use this information to answer the questions below. Between what values do the lengths of the middle 99.7% of all pregnancies fall? What percent of these pregnancies last more than 290 days? What percent of these pregnancies last between 258 and 290 days? How long is a pregnancy which falls into the percentile?

Example Suppose that the average height for adult males is normally distributed with a mean of 70 inches and a standard deviation of 2.5 inches. What percentile does a man who is 68 inches fall into? What percent of men are taller than 72 inches? How tall is a man in the 9.68 percentile? How tall is a man who has 8% of all men taller than him? Determine the percentage of men falling between inches and 73.5 inches.

Margin of Error (E or moe)
z* = is a critical value 90% z = 1.645 95% z = 1.96 99% z = 2.576 If you know a particular confidence level (%) and MOE, you can solve for your sample size, n.

Margin of Error (E or moe)
A smaller moe says that we have pinned down the parameter quite precisely. To make the margin of error smaller… make z* smaller make n bigger, which will cost more