AP Statistics HW: p.83 #1, 4, 6, 8 Obj: to understand density functions Do Now: Use your calculator to make a histogram for the following data on the height of cherry trees (in feet). Remember to adjust your window. You should have 6 bars. (Remember, you can control the # of bars by changing the “Xscl” for your window). Roughly sketch the graph with labels. C2 D

Draw a smooth curve through the top of the bars. You have drawn a density curve.  A mathematical model for the distribution  Preferable to a histogram because it can be described with a function.  Gives an overall picture of the dist’n while ignoring minor irregularities and outliers

Back to the histogram Shade the 1 st 3 bars. This shaded area represents 14 observations which are < 75. Since we have 31 observations, this is 14/31 or 45% of the data.

Now shade the area under your density curve to the left of 75. This area represents the probability that an observation will be less than 75. This area is considered to be.45 What is the area under the entire curve?

Ex: area = What does the density function tell you about the observations from this distribution? 2.What is the area under the entire curve?

Density Curve Any curve that is entirely above the x-axis and has an area under it that is equal to 1 could be a valid density function.

Normal Curve A normal curve is a special type of density function that is symmetric about the mean There are many different types of data that can be explained with a normal curve. A normal curve is sometimes referred to as a “bell curve”. This is a normal curve with a mean of 10:

Median and Mean of a density curve The median of the density curve is the equal- areas point, the point that divides the area under the curve in half The mean of the density curve is the balance point, at which the curve would balance if made of solid material. The median and mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean is a skewed curve is pulled away from the median in the direction of the long tail.

Mean of a density curve The point at which the curve would physically “balance”. Look at Figure 2.6 on p.82 Why is the mean to the right of the median?

Mean ( ) and standard deviation (s) are calculated from actual observations. Because the density curve is a “smoothed- over” version of the distribution, we use the following notation: mean: μ (mu) standard deviation: σ (sigma) Try question 2.2 on page 83