Download Section 5.1: Normal Distributions

Section 5.1: Normal Distributions – Finding Probabilities
To do at home
Read the book, listen to the assigned videos and do the WILEYPLUS HWK.
1) Go to WILEY HWK - Watch video on learning goal 1 – “Estimate probabilities as areas of a density function”.
a) What is a density curve?
b) What are the properties of a density curve?
2) Normal Distributions - Go to WILEY HWK - Watch video on learning goal 2 – Recognize how the mean and
standard deviation relate to the center and spread of a normal distribution.
a.
b.
c.
d.
e.
Sketch a normal density curve
What is its shape?
Give the parameters that define it
Give the notation used (it’s on the book)
Remember the empirical rule? State it here
i. About ________% of the distribution is within _______ standard deviation from the mean.
ii. About ________% of the distribution is within _______ standard deviation from the mean.
iii. About ________% of the distribution is within _______ standard deviation from the mean.
f.
Sketch three normal curves:
i. N(5, 1)
ii. N(12, 1)
iii. N(12, 2)
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Using the calculator to find probabilities – for WILEYPLUS instructions refer to the last page:
2nd VARS[DISTR], select 2: normalcdf(lower, upper, mean, standard deviation) ENTER
3) The Standard Normal Distribution: z ~ N (0, 1)
Find the specified areas for a N(0, 1) density. Sketch, label and shade.
(a)
(b)
(c)
The area below z = 0.8
The area above z = 1.2
The area between z = −1 and z = 1 – relate to Empirical Rule
4) Find areas under a normal curve. Sketch, label and shade
(a) The area above 6 on a N(5, 1.5) distribution
(b) The area below 15 on a N(20, 3) distribution
(c) The area between 90 and 100 on a N(100, 6) distribution
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5) Women’s heights are normally distributed with a mean of 65.0 inches and standard deviation of 3.5 inches.
Men’s heights have the same shape and parameters 70.0 inches and 4.0 inches. Use the calculator to find the
proportions.
a. Use correct notation to describe the distributions.
b. Sketch the corresponding density curves.
c. Find the proportion of women taller than 72 inches
d. Find the proportion of men shorter than 70 inches Is
e. Find the proportion of women with heights between 61.5 and 68.5 inches
f. Find the proportion of men with heights between 62 and 78 inches
Show work below, label a-f
6) 5.29 - What Proportion Have College Degrees?
According to the US Census Bureau,4 about 27.5% of US adults over the age of 25 have a bachelor's level (or
higher) college degree. For random samples of n=500 US adults over the age of 25, the sample proportions, ,
with at least a bachelor's degree follow a normal distribution with mean 0.275 and standard error 0.02.
a) Draw a sketch of this normal distribution and label at least three points on the horizontal axis.
b) How often will such samples have a proportion, , that is more than 0.30?
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Section 5.1 - Finding Scores from a given proportion
Using the calculator to scores, percentiles, endpoints of intervals – for WILEYPLUS instructions
refer to the last page:
2nd VARS[DISTR], select 3: Inv.Norm(Area to the left, mean, standard deviation) ENTER
7) Find endpoints on a N(0, 1) density curve.
1. What is z if the area to the left is about 0.10?
2. What is the z that separates the bottom 20% of the distribution from the top 80%?
3. What is the 90th percentile?
4. What is the score that separates the top 20% of the distribution?
5. What are the scores that have a 0.90 area between them?
6. What are the scores that have a 0.95 area between them?
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Converting from a general normal distribution N(mu, sigma) to the standard normal
distribution N (0, 1) and vice versa
8) Go to WILEY HWK – Watch video on learning goal 5 – Write the two formulas given in the video.
Using the calculator to scores, percentiles, endpoints of intervals – for WILEYPLUS instructions refer to
the last page:
2nd VARS[DISTR], select 3: Inv.Norm(Area to the left, mean, standard deviation) ENTER
9) Women’s and Men’s heights again - Women’s heights are normally distributed with a mean of 65.0 inches and
standard deviation of 3.5 inches. Men’s heights have the same shape and parameters 70.0 inches and 4.0
inches. Use the calculator to find the proportions.
a) Find the height in the 90th percentile for each distribution.
b) For each gender, find the height that separates the bottom 20% of the distribution.
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10) 5.37 - Quartiles for Commuting Times in St. Louis - Commuting times in St. Louis can be estimated using the
normal distribution N(21.97 minutes, 0.65 minutes). Find the first and third quartiles of the commuting times
distribution.
11) 5.34 - Boys Heights - Heights of 10-year-old boys (5th graders) follow an approximate normal distribution with
mean μ=55.5 inches and standard deviation σ=2.7 inches.6
(a) Draw a sketch of this normal distribution and label at least three points on the horizontal axis.
(b) According to this normal distribution, what proportion of 10-year-old boys are between 4 ft 4 in and 5 ft tall
(between 52 inches and 60 inches)?
(c) A parent says his 10-year-old son is in the 99th percentile in height. How tall is this boy?
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OPTIONAL
Using STATKEY to Find Normal Probabilities, proportions, areas under the normal curve
12) Go to WILEY HWK - Watch video on learning goal 3 – Using STATKEY to estimate proportions under a normal
curve. (In STATKEY – under THEORETICAL distributions, click on NORMAL)
a. How do you access it?
b. How do you change the x-value?
c. How do you change the proportion?
d. How do you change the parameters?
e. How do you find an area below, above or between?
Take notes of the examples and the answers; sketch and shade for each example. In class we are using the
calculator to accomplish the same.
13) Example from this video on learning goal 3: “birth weight of babies” - Copy the statement of the problem in this
paper and take notes.
a. Here you will learn how to use the standardized values (z-scores)
b. Also how to use the actual parameters given on the problem.
14) Go to WILEY HWK - Watch video on learning goal 4 – Using STATKEY to find endpoints of intervals with specified
probabilities. (In STATKEY – under THEORETICAL distributions, click on NORMAL)
Take notes of the examples and the answers; sketch and shade for each example. In class we are using
the calculator to accomplish the same.
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