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Normal Distribution

- Recall how we describe a distribution of

quantitative (continuous) data - plot the data (stemplot or histogram)
- look for the overall pattern (shape, peaks, gaps)

and departures from it (possible outliers) - calculate appropriate numerical measures of

center and spread (5-number summary and/or mean

s.d.) - then we may ask "can the distribution be

described by a specific model?" (one of the more

common models for symmetric, single-peaked

distributions is the normal distribution having a

certain mean and standard deviation) - can we imagine a density curve fitting fairly

closely over the histogram of the data? - a density curve is a curve that is always on or

above the horizontal axis (gt 0) and whose total

area under the curve is 1

- An important property of a density curve is that

areas under the curve correspond to relative

frequencies - see Figures 1.25a and 1.25b below. - rel. freq287/947.303

area .293 - Note the relative frequency of vocabulary scores

lt 6 is roughly equal to the area under the

density curve lt 6.

- We can describe the shape, center and spread of a

density curve in the same way we describe data

e.g., the median of a density curve is the

equal-areas point - the point on the horizontal

axis that divides the area under the density

curve into two equal (.5 each) parts. The mean

of the density curve is the balance point - the

point on the horizontal axis where the curve

would balance if it were made of a solid

material. (See figures 1.26b and 1.27 below)

- For a normal density curve we see the

characteristic bell-shaped, symmetric curve

with single peak (at the mean value ?) and spread

out according to the standard deviation (?) See

Figure 1.28 for a picture of ???and ??

- The 68-95-99.7 Rule describes the relationship

between ? and ?. See Figure 1.29 Go over

example 1.25-1.26 on page 59-61.

- How many different normal curves are there? Ans

One for every combination of values of ? and

?but they all are alike except for their ? and

?. So we take advantage of this and consider a

process called standardization to reduce all

normals to one we call the Standard Normal

Distribution. - Denote a normal distribution with mean ? and

standard deviation ? by N(?,?). Let X correspond

to the variable whose distribution is N(?,?).

We may standardize any value of X by subtracting

? and dividing by ? - this re-writes any normal

into a variable called Z whose values represent

the number of standard deviations X is away from

its mean. The standardized value is sometimes

called a z-score. - If X is N(?,?), then Z is N(0,1), where

Z(X-?)/?. - We can find areas under Z from Table A, and these

areas equal the corresponding areas under X.

- Consider Example 1.25. Let Xheight (inches) of

a young woman aged 18-24 years. Then X is

N(64.5", 2.5"). - What proportion of these women's heights are

between 62" and 67"? - What proportion are above 67"? Below 72"?
- What proportions of these women's heights are

between 61" and 66"? NOTE This cannot be

solved by the 68-95-99.7 rule - What proportion are below 64.5"? Below 68"?
- What proportion are between 58" and 60"?
- Etc., etc., etc. .
- What height represents the 90th percentile of

this aged woman? - All problems of this type are solvable by

sketching the picture, standardizing, and doing

appropriate arithmetic to get the final

answerthe last question above is what I call a

"backwards problem", since you're solving for an

X value while knowing an area

- Weve seen examples of data that seem to fit the

normal model, and examples of data that dont

seem to fit Because normality is an important

property of data for specific types of analyses

well do later, it is important to be able to

decide whether a dataset is normal or not. A

histogram is one way but a better graphical

method is through the normal quantile plot - A simple description of how to draw a normal

quantile plot is given on page 68 but for us,

a normal quantile plot is always going to be

drawn by software and it will allow us to assess

the normality of our data in the following sense - if the data points fall along the straight line

(and within the bands on the plot) then the data

can be treated as normal. Systematic deviations

from the line indicate non-normal distributions -

outliers often appear as points far away from the

pattern of the points... - the y-intercept of the line corresponds to the

mean of the normal distribution and the slope of

the line corresponds to the standard deviation of

the normal distribution

Normal quantile plot of CO2 Table 1.6 on page 26

Notice the systematic failure of the points to

fall on the line, especially at the low end where

the data is piled up. Also, note the outliers

at the high end Conclusion Not normal

Normal quantile plot of the IQ scores of 78 7th

grades students - Data in Table 1.9 on page 29

Notice that the data points follow the line

fairly well, though there is a slight curve at

the low-middle, indicating more data than would

be expected for a normal. The y-intercept is

around 110 (mean approx. 110) and the slope is

around 10 (s.d. is approx. 10). Conclusion

Normal

- Read section 1.3, paying careful attention to the

examples (especially 1.25-1.32). Work through

the examples yourself to make sure you understand

how they are done! - Work problems 1.108-1.110, 1.113 (applet),

1.114-1.117, 1.119, 1.120-1.139, 1.140-1.142,

1.143, 1.144, 1.148 - Try some of the Chapter 1 exercises (p. 78ff). By

test1 time, be sure you've worked as many of

the exercises in this chapter as you need to feel

comfortable with the material. - Don't forget the quizzes and homeworks on the

StatsPortal

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