Download Section 2 Normal Distributions

Section 2
Normal Distributions
Density Curves
Sometimes, the overall pattern of a large set of data is so regular that we describe it by a smooth curve.
The curve is a mathematical model, an idealized description (an approximation!), for the distribution,
ignoring minor irregularities and any outliers.
Density curves can have different shapes, but all have these same features:
 Describe overall pattern of a distribution
 Lie on or above the horizontal axis
 Have a total area equal to 1 under the curve
The area under a density curve gives the proportion of observations from the population that falls in that
region. We can determine the proportion of observations that lie within some range of values by finding
the area under the curve. (area = proportion = probability = percentile)
EXAMPLE:
a) Determine the height of the density curve.
b) What is the probability that a customer will take between 12 and 15 ounces of salad?
Uniform PDF for Salad Plate Filling Weight
Normal Distributions
Normal distributions have a symmetric, single-peaked, bell-shape, and are described by normal curves,
known by N(,  ) , where  represents the mean and  represents the standard deviation.
Empirical Rule (The 68-95-99.7 Rule)
Normal curves obey

 the following properties:

1. 68% of the observations fall within 1 𝜎 of the µ.
2. 95% of the observations fall within 2 𝜎 of the µ.
3. 99.7% of the observations fall within 3 𝜎 of the µ.
EXAMPLE: A radar unit is used to measure speeds of cars on a motorway. The speeds are normally
distributed with a mean of 90 km/hr and a standard deviation of 10 km/hr.
a. What percentage of cars has speeds between 70 and 110 km/hr?
b. What is the probability that a car picked at random is traveling at more than 100 km/hr?
c. What is the probability that a car picked at random is traveling at less than 70 km/hr?
Z-scores and standard deviations provide information about a value’s distance from the mean. A z-score
indicates the number of standard deviations above/below the mean that a particular value lies.
The Standard Normal Distribution is the normal distribution N(0, 1), where the mean is 0 and the
standard deviation is 1. With a standard normal distribution, where the standard deviation is equal to 1,
the number of standard deviations above/below the mean is equal to its z-score. For example, 2 standard
deviations below the mean is equal to a z-score of -2.
The Standard Normal Distribution has a population mean of 0 (   0 ) and a population
standard deviation of 1 (   1).


Notice that the area is the region under the curve, and the z scores represent distance along the
horizontal scale; the z-score is the number of standard deviations that a value is above or below
the mean.
For such standard normal distributions, we will use a table, in which:
 Cumulative area is given from the left up to a particular z-score.
 Z-scores are found in the left column and top row of the table.
 Areas are found in the body of the table.
Z-scores can be used to find percentiles (proportions, area, probability) given particular data values
(normalcdf) and to find particular data values given percentiles (invNorm). Percentiles represent the
ranking or position of a data value among all values of the population. An observation’s percentile is the
percent of the population that is at or to the left of that observation.
We can use a table to convert between z-scores and areas. (We can also use a graphing calculator.)
Finding Areas (proportions/percentiles) from Z-scores (from a Standard Normal Distribution)
1. Sketch a normal curve, label, and shade the region representing desired proportion.
2. Locate the given z-score in the table (from left column and top row) to find the area
(corresponding to cumulative from the left).
3. State the conclusion in terms of the problem.
EXAMPLES:
a. Find the proportion of observations from a standard normal distribution that are less than
-2.19.
b. Find the proportion of observations from a standard normal distribution that are greater than
1.43.
EXAMPLE: The temperatures of similar freezers produced by the FreezAir Company are approximated
by a standard normal distribution model, with a mean of 0oC and a standard deviation of 1.00oC. If one
freezer is randomly selected, find the probability that the temperature is less than 1.37o.
P(z a) [same as: P(z a)]
Sketch



Table/Work


Conclusion in context:
You can also find z-scores from given areas.
Finding Z-scores from Areas (proportions/percentiles) (from a Standard Normal Distribution)
1. Sketch a normal curve, label, and shade the region representing desired proportion.
2. Locate the given area in the table (in the body of the table) to find the z-score. (Select the
closest value or the largest value.)
3. State the conclusion in terms of the problem.
EXAMPLES:
a. What z-score corresponds to the 98th percentile in a standard normal distribution?
b. What x-value is the boundary for the top 10% in a standard normal distribution?
EXAMPLE: Using the same freezers, find temperature corresponding to the 85th percentile, P85 , the
temperature that separates the bottom 85% from the top 15%.
Sketch
Table/Work

Conclusion in context: About 85% of the freezers will have temperatures less than or equal to
and 15% of the freezers will have temperatures greater than or equal to
.
,
EXAMPLE: Using the same freezers, find the temperatures separating the bottom 2.5% and the top 2.5%.
Sketch
Table/Work
Conclusion in context:
With Nonstandard Normal Distributions, such as the radar unit example with N(90, 10), the mean is not
0 and/or the standard deviation is not 1; therefore, the standard deviation will not be equal to its z-score.
However, we can “standardize” the variable values into N(0, 1). When we standardize values with the
mean and standard deviation of a data set, we convert the data values (using the formula below) and
represent them as z-scores: z  x  

Standardizing values into z-scores:
—maintains theshape of the distribution
—changes the center to x = 0
—changes the spread to s = 1
The Nonstandard Normal Distribution does not have has a population mean of 0 and a
population standard deviation of 1, so we must convert values to standard scores using the
formula:
x 
(round z-scores to 2 decimal places)
z

The area bounded by some score x in any normal distribution is the same as the area
bounded by the equivalent z-score in the standard normal distribution:

Finding Areas (proportions/percentiles) from X-values (from a Non-Standard Normal Distribution)
1. Sketch a normal curve, label, and shade the region representing desired proportion.
x 
2. Convert the boundary x-value into a z-score, using: z 
 .
3. Locate that z-score in the table (from left column and top row) to find the area.
4. State the conclusion in terms of the problem.

EXAMPLE: The length of life of an instrument produced by a machine has a normal distribution with a
mean of 12 months and standard deviation of 2 months. Find the probability that an instrument produced
by this machine will last:
a. less than 7 months.
b. between 7 and 12 months
a) less than 7 months

b) between 7 & 12 months
1st: Sketch
1st: Sketch
2nd: Convert to z scores
P(x  7)
x
7 12
z

 2.5

2
z = -2.5
2nd: Convert to z scores
𝑃(7 < 𝑥 < 12)
x 
z


𝑧7 = −2.5 𝑎𝑛𝑑 𝑧12 = 0
3rd: Find probability using table
P(z  2.5)
4th: Conclusion in context:
About __________________ of the instruments will
last less than 7 months.

3rd: Find probability using table
P(z  0)  P(z  2.5)
4th: Conclusion in context:
EXAMPLES:
a. Find the percentile that corresponds to an ACT score of 30. (ACT scores N(18, 6))
b. Find the probability that a student scores at least 28 on the ACT.
c. Find the proportion of ACT scores that are between 16 and 20.
EXAMPLE: The Air Force had been using the ACES-II ejection seats designed for men weighing
between 140 lb. and 211 lb. Given that women’s weights are normally distributed with a mean of 143 lb.
and a standard deviation of 29 lb., what percentage of women have weights within those limits?
You can also find x-values from given areas.
Finding X-values from Areas (proportions/percentiles) (from a Non-Standard Normal Distribution)
1. Sketch a normal curve, label, and shade the region representing desired proportion.
2. Convert the given area (percentile) into z-scores using the table.
x 
3. Find boundary x-values using the obtained z-scores and z 
 .
4. State the conclusion in terms of the problem.

EXAMPLE: In designing seats to be installed in commercial aircraft, engineers want to make the seats
wide enough to fit 98% of all males. Men have hip breadths that are normally distributed with a mean of
14.4 in. and a standard deviation of 1.0 in. Find P98 .

Standardizing values into z-scores also allows us to compare among different data sets with different
scales.
EXAMPLE:
Mark scores a 680 on the mathematics part of the SAT. The distribution of SAT scores in a reference
population is normal, with mean 500 and standard deviation 100. Gabby takes the American College
Testing (ACT) mathematics test and scores 27. ACT scores are normally distributed with mean 18 and
standard deviation 6. Find the standardized scores for both students. Assuming that both tests measure
the same kind of ability, who has the higher score?
Later, we will be required to determine that the data come from populations that are approximately
normally distributed. We can observe the shape of histograms, stemplots, and boxplots to see how well
the data fit the 68-95-99.7 rule for normal distributions (though some non-normal distributions fit the rule
too), or we can look at a Normal Probability Plot to verify such. When looking at a normal probability
plot, a straight diagonal line indicates a roughly normal distribution while a bent, curved line indicates
skewness.
Normal distribution
Right-skewed distribution
PRACTICE:
a. How tall does a woman have to be to be 3 standard deviations above the mean? (Heights of
young women N(64.5, 2.5))
b. Find P98 of women’s heights.

c. What is the minimum height of a woman who is in the top 10%?
d. Draw a normal probability plot to determine if the following data sets represent normal
distributions.
i. 6 7 7 12 5 8 6 8 21
ii. 5
9
3
1
3
5
5
1
5
5
5
5
8
8
3