Finish Section 2.1

Density Curves

1. Always plot your data: dotplot, stemplot, histogram…2. Look for overall pattern (SOCS)3. Calculate numerical summary to describeAdd a step…4. Sometimes the overall pattern is so regular we can describe

it by a smooth curve.

A Density Curve is a curve that:• Is always on or above the horizontal axis• Has an area of 1 underneath it

Things to note…• The mean is the physical balance point of a density curve or a

histogram

• The median is where the areas on both sides of it are equal

Don’t forget…• The mean and the median are equal for symmetric density

curves!

Start Section 2.2Norman Curves!

Normal Distributions = Normal Curves

• Any particular Normal distribution is completely specified by two numbers: its mean () and standard deviation ().

• We abbreviate the Normal distribution with mean and standard deviation as N()

The 68-95-99.7 Rule

Other images to explain the same thing…in case it helps!

Example• A pair of running shoes lasts on average 450 miles, with a

standard deviation of 50 miles. Use the 68-95-99.7 rule to find the probability that a new pair of running shoes will have the following lifespans.

Between 400-500 miles

More than 550 miles

Example• A survey of 1000 U.S. gas stations found that the price charged for a

gallon of regular gas can be closely approximated by a normal distribution with a mean of $1.90 and a standard deviation of $0.20. How many of the stations charge:

a. between $1.50 and $2.30 for a gallon of regular gas?

b. less than $2.10 for a gallon of regular gas?

c. more than $2.30 for a gallon of regular gas?

Example• A vegetable distributor knows that during the month of August, the

weights of its tomatoes were normally distributed with a mean of 0.61 pound and a standard deviation of 0.15 pound.

a. What percent of the tomatoes weighed less than 0.76 pound?

b. In a shipment of 6000 tomatoes, how many tomatoes can be expected to weigh more than 0.31 pound?

c. In a shipment of 4500 tomatoes, how many tomatoes can be expected to weigh between 0.31 and 0.91 pound?

How does this relate to z-scores?

• If the distribution happens to be normal we can find the area under the curve and percentiles by using z-scores!

• Standard Normal distribution – mean of 0 and standard deviation of 1.

Table A in your book…• The table entry for each z-score is the area under the curve to

the left of z.

• If we wanted the area to the right, we would have to subtract from 1 or 100%

Examples• Find the proportion of observations that are less than 0.81

• Find the proportion of observations that are greater than -1.78

• Find the proportion of observations that are less than 2.005

• Find the proportion of observations that are greater than 1.53

• Find the proportion of observations that are between -1.25 and 0.81

Homework• Pg 107 (19-24, 27, 28)