《统计学基础(英文版·第7版)》教学课件les7e-05-01

1、ChapterNormal Probability Distributions5ChapterNormal Probability DistChapter OutlineChapter OutlineSection 5.1Introduction to Normal Distributions and the Standard Normal DistributionsSection 5.1Introduction to NorSection 5.1 ObjectivesHow to interpret graphs of normal probability distributionsHow

2、to find areas under the standard normal curveSection 5.1 ObjectivesHow to iProperties of a Normal DistributionContinuous random variable Has an infinite number of possible values that can be represented by an interval on the number line.Continuous probability distributionThe probability distribution

3、 of a continuous random variable.Hours spent studying in a day06391512182421The time spent studying can be any number between 0 and 24.Properties of a Normal DistribProperties of a Normal Distribution.Normal distribution A continuous probability distribution for a random variable, x. The most import

4、ant continuous probability distribution in statistics.The graph of a normal distribution is called the normal curve. xProperties of a Normal DistribProperties of a Normal Distribution.The mean, median, and mode are equal.The normal curve is bell-shaped and is symmetric about the mean.The total area

5、under the normal curve is equal to one.The normal curve approaches, but never touches the x-axis as it extends farther and farther away from the mean.xTotal area = 1Properties of a Normal DistribProperties of a Normal Distribution.Between and + (in the center of the curve), the graph curves downward

6、. The graph curves upward to the left of and to the right of + . The points at which the curve changes from curving upward to curving downward are called the inflection points. Inflection points 3 + 2 + 2 + 3xProperties of a Normal DistribProbability Density Function (PDF).A discrete probability dis

7、tribution can be graphed with a histogram. For a continuous probability distribution, you can use a probability density function (pdf). A probability density function has two requirements:the total area under the curve is equal to 1 the function can never be negative. Probability Density Function (M

8、eans and Standard DeviationsA normal distribution can have any mean and any positive standard deviation.The mean gives the location of the line of symmetry.The standard deviation describes the spread of the data.Means and Standard DeviationsAExample: Understanding Mean and Standard Deviation.Which c

9、urve has the greater mean?Solution:Curve A has the greater mean (The line of symmetry of curve A occurs at x = 15. The line of symmetry of curve B occurs at x = 12.)Example: Understanding Mean anExample: Understanding Mean and Standard Deviation.Which curve has the greater standard deviation?Solutio

10、n:Curve B has the greater standard deviation (Curve B is more spread out than curve A.)Example: Understanding Mean anExample: Interpreting Graphs of Normal Distributions.The scaled test scores for New York State Grade 4 Common Core Mathematics Test are normally distributed. The normal curve shown be

11、low represents this distribution. What is the mean test score? Estimate the standard deviation of this normal distribution. (Adapted from New York State Education Department)Example: Interpreting Graphs oSolution:The scaled test scores for the New York State Grade 4 Common Core Mathematics Test are

12、normally distributed with a mean of about 305 and a standard deviation of about 40.Solution: Interpreting Graphs of Normal Distributions.Solution:Solution: InterpretinSolution:Using the Empirical, you know that about 68% of the scores are between 265 and 345, about 95% of the scores are between 225

13、and 385, and about 99.7% of the scores are between 185 and 425.Solution: Interpreting Graphs of Normal Distributions.Solution:Solution: InterpretinThe Standard Normal Distribution.Standard normal distribution A normal distribution with a mean of 0 and a standard deviation of 1.Any x-value can be tra

14、nsformed into a z-score by using the formula 3121023 zArea = 1The Standard Normal DistributiThe Standard Normal Distribution.The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The total area under its normal curve is 1.The Standard Normal Distri

15、butiProperties of the Standard Normal Distribution.The cumulative area is close to 0 for z-scores close to z = 3.49.The cumulative area increases as the z-scores increase.z3121023z = 3.49Area is close to 0Properties of the Standard NorProperties of the Standard Normal Distribution.The cumulative are

16、a for z = 0 is 0.5000.The cumulative area is close to 1 for z-scores close to z = 3.49.z = 3.49Area is close to 1Area is 0.5000z = 0z3121023Properties of the Standard NorExample: Using The Standard Normal Table.Find the cumulative area that corresponds to a z-score of 1.15.The area to the left of z

17、= 1.15 is 0.8749.Move across the row to the column under 0.05Solution:Find 1.1 in the left hand column.Example: Using The Standard NoExample: Using The Standard Normal Table.Find the cumulative area that corresponds to a z-score of 0.24.Solution:Find 0.2 in the left hand column.Move across the row t

18、o the column under 0.04.The area to the left of z = 0.24 is 0.4052.Example: Using The Standard NoFinding Areas Under the Standard Normal Curve.Sketch the standard normal curve and shade the appropriate area under the curve.Find the area by following the directions for each case shown.To find the are

19、a to the left of z, find the area that corresponds to z in the Standard Normal Table.The area to the left of z = 1.23 is 0.8907Use the table to find the area for the z-scoreFinding Areas Under the StandaFinding Areas Under the Standard Normal Curve.To find the area to the right of z, use the Standar

20、d Normal Table to find the area that corresponds to z. Then subtract the area from 1.Subtract to find the area to the right of z = 1.23: 1 0.8907 = 0.1093.The area to the left of z = 1.23 is 0.8907.Use the table to find the area for the z-score.Finding Areas Under the StandaFinding Areas Under the S

21、tandard Normal Curve.To find the area between two z-scores, find the area corresponding to each z-score in the Standard Normal Table. Then subtract the smaller area from the larger area.Subtract to find the area of the region between the two z-scores: 0.8907 0.2266 = 0.6641.The area to the left of z = 0.75 is 0.2266.The area to the left of z = 1.23 is 0.8907.Use the table to find the area for the z-scores.Finding Areas Under the StandaExample: Finding Area Under the