1、ChapterNormal Probability Distributions5Chapter OutlineSection 5.1Introduction to Normal Distributions and the Standard Normal DistributionsSection 5.1 Objectives How to interpret graphs of normal probability distributions How to find areas under the standard normal curveProperties of a Normal Distr
2、ibutionContinuous random variable Has an infinite number of possible values that can be represented by an interval on the number line.Continuous probability distribution The probability distribution of a continuous random variable.Hours spent studying in a day06391512182421The time spent studying ca
3、n be any number between 0 and 24.Properties of a Normal Distribution.Normal distribution A continuous probability distribution for a random variable,x.The most important continuous probability distribution in statistics.The graph of a normal distribution is called the normal curve.xProperties of a N
4、ormal Distribution.1.The mean,median,and mode are equal.2.The normal curve is bell-shaped and is symmetric about the mean.3.The total area under the normal curve is equal to one.4.The normal curve approaches,but never touches the x-axis as it extends farther and farther away from the mean.xTotal are
5、a=1Properties of a Normal Distribution.5.Between and +(in the center of the curve),the graph curves downward.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
6、+2 +2+3xProbability Density Function(PDF).A discrete probability distribution 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:1.the total area under the curve is equal to 1 2
7、.the function can never be negative.Means and Standard Deviations A 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.Example:Understanding Mean and Standard Deviation.
8、1.Which curve 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 and Standard Deviation.2.Which curve has the greater standard deviation?Solution:Curve B has the greater
9、 standard deviation(Curve B is more spread out than curve A.)Example: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 below represents this distribution.What is the mean test sco
10、re?Estimate the standard deviation of this normal distribution.(Adapted from New York State Education Department)Solution:The scaled test scores for the New York State Grade 4 Common Core Mathematics Test are normally distributed with a mean of about 305 and a standard deviation of about 40.Solution
11、:Interpreting Graphs of Normal Distributions.Solution:Using the Empirical,you know that about 68%of the scores are between 265 and 345,about 95%of the scores are between 225 and 385,and about 99.7%of the scores are between 185 and 425.Solution:Interpreting Graphs of Normal Distributions.The Standard
12、 Normal Distribution.Standard normal distribution A normal distribution with a mean of 0 and a standard deviation of 1.Any x-value can be transformed into a z-score by using the formula ValueMeanStandard deviationxz3121023 zArea=1The Standard Normal Distribution.The standard normal distribution is a
13、 normal distribution with a mean of 0 and a standard deviation of 1.The total area under its normal curve is 1.Properties of the Standard Normal Distribution.1.The cumulative area is close to 0 for z-scores close to z=3.49.2.The cumulative area increases as the z-scores increase.z3121023z=3.49Area i
14、s close to 0Properties of the Standard Normal Distribution.3.The cumulative area for z=0 is 0.5000.4.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=0z3121023Example:Using The Standard Normal Table.1.Find the cumulative area that corresponds to
15、a z-score of 1.15.The area to the left of z=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 Normal Table.2.Find the cumulative area that corresponds to a z-score of 0.24.Solution:Find 0.2 in the left hand column.Move acr
16、oss the row to the column under 0.04.The area to the left of z=0.24 is 0.4052.Finding Areas Under the Standard Normal Curve.1.Sketch the standard normal curve and shade the appropriate area under the curve.2.Find the area by following the directions for each case shown.a.To find the area to the left
17、 of z,find the area that corresponds to z in the Standard Normal Table.2.The area to the left of z=1.23 is 0.89071.Use the table to find the area for the z-scoreFinding Areas Under the Standard Normal Curve.b.To find the area to the right of z,use the Standard Normal Table to find the area that corr
18、esponds to z.Then subtract the area from 1.3.Subtract to find the area to the right of z=1.23:1 0.8907 =0.1093.2.The area to the left of z=1.23 is 0.8907.1.Use the table to find the area for the z-score.Finding Areas Under the Standard Normal Curve.c.To find the area between two z-scores,find the ar
19、ea corresponding to each z-score in the Standard Normal Table.Then subtract the smaller area from the larger area.4.Subtract to find the area of the region between the two z-scores:0.8907 0.2266=0.6641.3.The area to the left of z=0.75 is 0.2266.2.The area to the left of z=1.23 is 0.8907.1.Use the ta
20、ble to find the area for the z-scores.Example:Finding Area Under the Standard Normal Curve.Find the area under the standard normal curve to the left of z=0.99.From the Standard Normal Table,the area is equal to 0.1611.Solution:Example:Finding Area Under the Standard Normal Curve.Find the area under
21、the standard normal curve to the right of z=1.06.From the Standard Normal Table,the area is equal to 0.1446.Solution:Example:Finding Area Under the Standard Normal Curve.Find the area under the standard normal curve between z=1.5 and z=1.25.From the Standard Normal Table,the area is equal to 0.8276.So,82.76%of the area under the curve falls between z=1.5 and z=1.25.Solution:
