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Measuring Location:z-scores
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Example 1Distribution of head circumference for malesoldiers is
approximately normal with mean = 22.8 inches and std. dev. 1.1
inches.
a) What percent of soldier have head circumference greater than
23.9 inches?
b) Between 21.7 and 23.9 inches?
c) Less than 25.1 inches?
UH OH…..what do we do now?????!?!?!?
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Standardizing and Z-Scores● All normal distributions are the
same if we measure in
units of size σ about the mean μ as center. ● Z-score: x is an
observation that has μ and σ, so the
standardized value is:
● Always round the z-score to 2 decimal places!
x - meanz =
standard deviation
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Standardizing and Z-Scores● The z-score tells us how many
standard deviations the
original observation falls away from the mean.
○ If x > μ, the z-score will be positive.
○ If x < μ, the z-score will be negative.
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Example 1, continued...● Distribution of head circumference for
male soldiers is
approximately normal with mean = 22.8 inches and std. dev. 1.1
inches.
● Find the z-score for head sizes less than 25.1 in
x - meanz =
standard deviation
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Standard Normal Distribution● N(0, 1)● Find the area under any
standard normal curve using
Table A or the z- table.
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Example 1, almost finished...Distribution of head circumference
for male soldiers is approximately normal with mean = 22.8 inches
and std. dev. 1.1 inches.
The z-score for head size less than 25.1 in is 2.09.
Using Table A find the probability of head sizes less than 25.1
inches.
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Example 2Women’s heights are normally distributed with μ = 64.5
inches and σ = 2.5 inches.
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Example 2● What proportion of all young women are less than
68
inches tall?
● Find the z-score for height less than 68 in.
x - meanz =
standard deviation
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Example 2, continued...Understand that the area under the
N(64.5, 2.5) curve to the left of x = 68 is equal to the area to
the left of z = 1.4 under the standard normal curve N(0, 1).
Use Table A to find the percent of women that are lessthan 68
inches tall.
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Remember!The z- value in Table A is the area/percentage to the
left of the z-score.
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Examples with Table A1. Find the proportion of observations from
the standard
normal distribution that are less than 3.3
2. Find P (Z > -2.15)
3. Find the proportion of observations from the standard normal
distribution that are between -2.15 and 1.4.
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Examples with Table A4. Find the proportion of observations from
the standard
normal distribution that are less than 2.2.
5. Find the proportion of observations from the standard normal
distribution that are greater than -1.35.
6. P(-1.30 < Z < 2.65)
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Standardizing and Z-Scores● We can also use z-scores to compare
the locations of
individuals in different distributions.
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FINDING NORMAL PROPORTIONS &
FINDING A VALUE GIVEN A PROPORTION
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Finding Normal Proportions1. Standardize x to a standard normal
variable z.
2. Draw a picture and shade the area of interest under the
standard normal curve.
3. Find area using Table A and the fact that the total area
under the curve is 1.
4. Write a conclusion in context of the problem.
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Example 1Height for women is N(64.5, 2.5).
a. Find P (x< 68)
b. Find P (x > 63)
c. Find P (63 < x < 68)
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Example 2For 14-year old boys the distribution of blood
cholesterol levels is normal with N(170, 30).
a. What percent of 14 year old boys have more than 240 mg/dl of
cholesterol?
b. P (x ≤ 225)
c. P( 170 ≤ x ≤ 240)
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Example 3
Jessica took the SATs and the ACTs. On the SATs she got a 610.
The SATs are normally distributed with a mean of 500 and standard
deviation of 100. On the ACTs she got a 27. The ACTs are normally
distributed with a mean of 18 and standard deviation of 6. On which
test did she get a better score?
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Finding a Z-Value Given a Proportion(aka Working Backwards with
Table A)Example 4Find the value z on the standard normal curve that
is less than only the top 30% of the data.
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Finding a Z-Value Given a Proportion(aka Working Backwards with
Table A)
Example 5Find the value on the curve X = N(60, 2) that is
greater than 20% of the population.
