Download Block: ______ Chapter 3: Modeling Distribution

 Statistics Name: ___________________________
Teacher: _______________
Block: ______ Chapter 3: Modeling Distribution of Data Section 3.1: Density Curves ●
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This is a histogram of the scores of 947 seventh­grade students on a standardized vocabulary test. Scores on this test have a very ___________ __________________. Describe the histogram. ●
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Shape: _____________________________ Peak: ______________________________ Gaps: ______________________________ Outliers: ____________________________ The ________________________ drawn through the tops of the histogram bar is a good description of the _______________________________ of the data. You can think of drawing a ______________ through the ___________ of the histogram and smoothing out the ________________________________________________ of the bars. ●
This is called a ________________________________. Density Curves One important distinction between histograms and density curves is most histograms show the _____________ of observations in each ___________ by the ______________ of their ______ and therefore by the _____________________________. We set up density curves to show the ____________________________ of observations in any ________________ by the area ________________________________. We do this by choosing a ________________ so that the _____________________ under the curve is exactly _______. Statistics Name: ___________________________
Teacher: _______________
Block: ______ Why we use Density Curves: ●
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Histograms give us more information that we need to analyze our data. The density curve is a model of a histogram with an _____________________________ __________________________ and ________________________________________. Characteristics of Density Curves ●
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A density curve is always _________________________________________________. The ______________________ of a density curve is always in ___________________. The _____________________ under any density curve is _______________________. Represents _________________________________________ Percentiles The ________ percentile of a distribution is _________________________________________ ____________________________________________________________________________. To find _____________________ from a density curve, we will find _____________________ _______________________________________________________________. First Graph The area of the _______________________________ represents the students with vocabulary scores _______________________________. There are 287 such students, who make up the _____________________________________. Therefore, a score of _______ corresponds to about the ______________________________. Statistics Name: ___________________________
Teacher: _______________
Block: ______ Second Graph Adjust the ______________ of the graph so that the ____________________________________________ is exactly ______. Areas under the curve then represent _______________ of the observations. ●
The curve is now a density curve. The ____________________________ under the density curve represents the ___________________________ of students with scores _____________________________. This area is __________, only _________________________________________________. This ___________________ corresponds to about the ______________________________. You can see that this is ________________________________________________________ ___________________________________________________________________________. Center and Spread of a Density Curve ●
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The _______________ is the point with ___________ the observations on either side. ○ The _______________ of a density curve is the _________________________. ○ This is the point with _______________________________________________ and ____________________________________________________________. The ________________ divide the area under the curve in ______________________. ○ ________________ of the area under the curve is ________________________ _________________________. ○ _________________ of the area is ___________________________________. You can roughly ________________________________________________________ of any density curve by ___________________________________________________ ______________________________________________________________________. The _________ of a density curve is the _____________________________. ○ It is the point at which ______________________________________________ ________________________________________________________________. The mean and median of a ______________________________________________________. They both ___________________________________________________________________. The mean of a _______________________________________________________________. Statistics Name: ___________________________
Teacher: _______________
Block: ______ If the density curve is _________________________, the mean is equal to the median. The median _____________________________. The mean is the ___________________________; where the curve would ______________________________________. The mean is ______________________________________. New Symbols ______
represents the mean of a density curve (pronounced ‘mu’) ______
represents the sample population. ______
represents the standard deviation of a density curve (pronounced ‘sigma’) Example 1 a.
Why is the total area under this curve equal to 1? b.
What percent of the observations lie above 0.8? c.
What percent of the observations lie below 0.6? d.
What percent of the observations lie between 0.25 and 0.75? e.
What is the mean, μ of this distribution? Statistics Name: ___________________________
Teacher: _______________
Block: ______ Example 2 An unusual ‘broken line’ density curve. Use areas under this density curve to find the proportion of observations within the given interval. a.
0.6 ≤ x ≤ 0.8 b.
0 ≤ x ≤ 0.4 c.
0 ≤ x ≤ 0.2 d.
The median of this density curve is a point between x = 0.2 and x = 0.4. Explain why. Example 3 At which of these points on each curve do the mean and the median fall? Example 4 Consider a uniform distribution that outputs numbers between 0 and 2. a.
Find P (x < 0.5) b.
Find P (x > 0.7) c.
Find P (0.2 < x < 1.2)