Chapter 2: Modeling Distributions of Data



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Chapter 2: Modeling Distributions of Data

2.2A: Describing a Location in a Distribution

( Density Curve

A density curve is a curve that:

• is always on or above the _________________ axis, and

• has area exactly ___ underneath it.

A density curve describes the __________ pattern of a distribution. The ______ under the curve and above any interval of values on the horizontal axis is the ________________ of all observations that fall in that interval.

Mean vs. Median in a Density Curve

Mean: still the balancing point( Median: still the equal-areas point.

Sketch the mean and median for each of the following graphs…

Chapter 2: Modeling Distributions of Data

2.2: Normal Distributions

( Normal Distributions

One particularly important class of density curves are the Normal curves, which describe Normal distributions.

• All Normal curves are ___________________, single-peaked, and ________-shaped.

• A specific Normal curve is described by giving its mean ___ and standard deviation ____.

Definition:

A Normal distribution is described by a Normal density curve. Any particular normal distribution is completely specified by two numbers: its mean ( and standard deviation σ.

• The mean of a Normal distribution is the __________ of the symmetric normal curve.

• The standard deviation is the distance from the center to the _________-of-__________ points on either side.

• We abbreviate the normal distribution with mean ( and standard deviation σ as N((, σ).

( The 68-95-99.7 Rule (“The Empirical Rule”)

Although there are many Normal curves, they all have properties in common.

Definition:

In the Normal distribution with mean ( and standard deviation σ:

• Approximately 68% of the observations fall within σ of (.

• Approximately 95% of the observations fall within 2σ of (..

• Approximately 99.7% of the observations fall within 3σ of (.

Helmet Sizes:

The army reports that the distribution of head circumference among male soldiers is approximately normal with mean 22.8 inches and standard deviation 1.1 inches, (that is, N(22.8, 1.1)).

• What percent of soldiers have head circumference between 21.7 inches and 23.9 inches?

• What percent of soldiers have a head circumference greater than 23.9 inches?

• What percent of soldiers have a head circumference less than 20.6 inches?

• Between what head circumferences do the middle 95% of soldiers fall?

( The Standard Normal Distribution

All Normal distributions are the same if we measure in units of size σ from the mean ( as center.

Definition:

The standard Normal distribution is the Normal distribution with mean ___ and standard deviation ___. If a variable x has any Normal distribution N((,σ) with mean ( and standard deviation σ, then the standardized variable

z =

has the standard Normal distribution, N(0, 1).

( The Standard Normal Table

Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table.

Definition:

Table A is a table of areas under the standard Normal curve. The table entry for each value z is the ________ under the curve to the _______ of z.

Use Table A to find:

• z < –2.2

• z > 1.34

• z > 5.2

• 0.58 < z < 1.79

• Find z that corresponds with the 20th percentile.

• Find z such that 45% of all observations are greater than z.

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