Download Ch. 2 - Normal Distributions - Stahler - Statistics

Ch. 2 - Normal Distributions - Stahler - Statistics - Page 1
Essential Question(s): How do we model density curves and normal distributions?
The following dotplots display scores in the first 2 rounds of the 2011 US Open.
How would you describe the overall pattern? Draw the curve (mathematical model) for each distribution?
* Mathematical models ignore minor irregularities as well as outliers.
Density Curve: area under the curve represents proportions of observations (total area under the curve = 1).
Graph is always above or on the x-axis.
The density curves above are approx. normal. Curves can also be skewed left and right (area still = 1).
Draw a normal, left skewed, and right skewed density curve. Approximate the mean and median in each curve.
How much of the area is to the right/left of the median? Quartiles?
*The mean is the point at which the curve would balance if it were made of solid material.
EX: Given the following curve, complete the following.
1) Verify that the graph is a valid density curve.
2) Find the area between the interval
3) Explain why the median is between 0.2 and 0.4
4) Find the area between the interval
5) Find the area between the interval
Ch. 2 - Normal Distributions - Stahler - Statistics - Page 2
Essential Question(s): How do we model density curves and normal distributions?
Enter the following command on your Ti-83/84. RandInt(1,6,200)->L1 (RandInt is found under Math->PRB)
What just happened? Turn Plot 1 on and select Histogram. Adjust your window to accommodate our
outcomes and graph.
1) Describe what the graph looks like and why it looks that way. 2) What would happen if 200 was 300?
500?
Outcome – the result of a single trial: 1, 2, 3, 4, 5, 6
Sample Space – set of all possible outcomes: {1, 2, 3, 4, 5, 6}
Event – a subset of the sample space: odds, primes, evens, less than 2,...
Enter the following command on your Ti-83/84. RandInt(1,6,200)->L1 & RandInt(1,6,200)->L2 . Find the sum
of both lists and store this value in L3. What does L3 represent? Graph (make sure to select L3).
Explain why this simulation created a frequency histogram whose density curve would be described as a
normal distribution.
Properties of Normal Distributions - Normal Distribution in the Real World
1) Bell-Shaped
2) Mean and median are located in the center
3) Symmetry about the mean/median
4) The larger the standard deviation, the flatter the curve (the opposite can be said about a small st. dev.)
Inflection Points: the points at which change of curvature take place
What is so great about normal distributions?
1) Good descriptions for some distributions of real data (grades, SAT scores, batting averages,...)
2) Good approximations to the results of many kinds of chance outcomes (flipping coins, dice,...)
3) Many statistical inference procedures based on normal dist. work well for other roughly symmetric dist.
Empirical Rule: all normal distributions obey the rule that:
68% of observations fall in the range of
95% of observations fall in the range of
99.7% of observations fall in the range of
Ex: The dist. of heights of women (18-24) is approximately normal with
1) Find the range of values in which 95% of women’s heights are between.
2) How tall would you be if you were in the 16th percentile?
and
.
HW: 1) The distribution of heights of adult American men is approximately normal with mean 69 inches and st.
dev. 2.5 inches. Draw a normal curve on which this mean and standard deviation are correctly located. Use the
Empirical Rule (68-95-99.7) to answer the following.
a) What percent of men are taller than 74 inches?
b) Between what heights do the middle 95% of men fall?
c) A height of 71.5 inches corresponds to what percentile of adult male American heights?
2) The army reports that the distribution of head circumferences among male soldiers is approximately normal
with mean 22.8 inches and st. dev. 1.1 inches. Use the Empirical Rule (68-95-99.7) to answer the following.
a) What percent of soldiers have head circumference greater than 20.6 inches?
b) A head circumference of 25 inches would be what percentile?
c) What percent of soldiers have head circumference between 21.7 inches and 23.9 inches?
3) The length of human pregnancies from conception to birth varies according to a distribution that is
approximately normal with mean 266 days and standard deviation 16 days. Use the Empirical Rule (68-9599.7) to answer the following.
a) Between what values do the lengths of the middle 95% of all pregnancies fall?
b) How short are the shortest 2.5% of all pregnancies?
c) How long are the longest 2.5% of all pregnancies?
4) Wechsler Adult Intelligence Scale (WAIS) IQ scores for young adults are N(110, 25). *N(110, 25) is just an
abbreviation for a normal distribution “N” with
a) If someone’s score were reported at the 16th percentile, about what score would that individual have?
b) Answer the same question for the 84th and 97.5th percentile.
Ch. 2 - Normal Distributions - Stahler - Statistics - Page 4
Essential Question(s): How do we approximate the area under the curve of a normal distribution?
The Empirical Rule is a great tool if our data falls 1, 2, or 3 standard deviations above or below the mean. What
if the numbers don’t work so nicely?
Z-Score (standardized value): The number of st. dev. an observation (x) is above or below the mean.
*z-scores can be + or Ex: The heights of young women are approximately normal with
. How
many standard deviations above the mean is a woman that is 68 inches tall? 59.5 inches? 64.5 inches?
Standard Normal Distribution: a normal dist. N(0,1) with mean 0 and standard deviation 1.
Standard normal dist. give us a common scale which enables us to compare data from different observations.
Ex: Three landmarks of baseball achievements are Ty Cobb’s batting average of .420 in 1911, Ted William’s
.406 in 1941, and George Brett’s .390 in 1980. These batting averages cannot be compared directly because
the distribution of MLB batting averages has changed over the years. The distributions are quite symmetric and
reasonably normal. While the mean batting average has been held roughly constant by rule changes and the
balance between hitting and pitching, the standard deviation has dropped over time.
a) What does it mean that standard deviations have dropped over time?
b) Compute the z-scores (standardized) for Cobb, Williams, and Brett to compare how far each stood above
his peers.
Cobb
Williams
Brett
Decade
Mean
St. Dev.
1910s
.266
.0371
1940s
.267
.0326
1970s
.261
.0317
Since this baseball data is standardized, we can find the area under any normal curve (percentile) provided we
know the z-score.
YouTube Video - How to find the area under the standard normal curve given a z-score.
Ex: Find the area under the curve for the following z-scores. Draw a picture and shade the area of interest.
a)
b)
c)
d)
e)
HW: Find z and draw a picture of the shaded area. Find the area.
1) N(45,2.1) area left of x=47
2) N(12,0.45) area right of x=13.2
25
3) N(22,1.1) area between 23 and
Enter the following into your calculator: normalcdf(-99,.95), normalcdf(2.67,99), normalcdf(.91,2.73)
*normalcdf can be found under 2nd, VARS
4) The level of cholesterol in the blood is important because high cholesterol levels may increase the risk of
heart disease. The distribution of blood cholesterol levels in a large population of people of the same age and
sex is roughly normal. For 14-year-old boys, the mean is
milligrams of cholesterol per deciliter of blood
(mg/dl) and the standard deviation is
mg/dl. Levels above 240 mg/dl may require medical attention.
a) What percent of 14-year-old boys have more than 240 mg/dl of cholesterol? Draw a picture and explain your
conclusion in the context of the problem.
b) What percent of 14-year-old boys have less than 200 mg/dl of cholesterol (considered normal)? Draw a
picture and explain your conclusion in the context of the problem.
c) What percent of 14-year-old boys have between 200 and 239 mg/dl of cholesterol (considered borderline
high)? Draw a picture and explain your conclusion in the context of the problem.
5) For a sample of 42 rabbits, the mean weight is 5 pounds and the standard deviation of weights is 3 pounds.
Which of the following is most likely true about the weights for the rabbits in this sample (2012)?
a) The distribution of weights is approximately normal because the standard deviation is less than the mean.
b) The dist. of weights is skewed right because the least possible weight is within 2 stdev.of the mean.
c) The dist. of weights is skewed left because the least possible weight is within 2 stdev of the mean.
d) The distribution of weights has a median that is greater than the mean.
Ch. 2 - Normal Distributions - Stahler - AP Statistics - Page 6
Essential Question(s): Given a %, how do we find a an x-value?
We know that
. Solve this equation for x.
Scores on the SAT verbal test follow approximately N(505,110) distribution. What score would you need to be
in the 90th percentile?
What score would you need to be in the top 25% of all students taking the SAT?
Between what scores would you need to be in the middle 50% of all students taking the SAT?
How can you assess normality?
Create 200 random integers from 1-6 and store them in L1. Repeat for L2.
Find the sum of these two lists and store the values in L3.
Create a histogram (Stat Plot). What should it look like if it were to be considered normal?
Create a boxplot. What should it look like if it were to be considered normal?
Create a normal probability plot (last selection). Use Zoom-Stat to automatically adjust the window. In this
graph, the x-values represent the values in L3 and the y-values are the corresponding z-scores. What should it
look like if it were to be considered normal?
HW: Draw pictures and explain your solution in the context of the problem.
1) A survey indicates that for each trip to the supermarket, a shopper spends an average of 45 minutes with a
standard deviation of 12 minutes in the store. The length of time spent in the store is normally distributed and is
represented by the variable x.
a) What % of shoppers were in the store less than 25 minutes?
b) What % of shoppers were in the store more than 39 minutes?
c) What % of shoppers were in the store between 24 and 54 minutes? If 200 shoppers enter the store, how
many of them would be there between 24 and 54 minutes?
d) If you wanted to find the time (to the nearest second) the fastest 5% complete their shopping in, what would
that time be?
e) If you wanted to find the time (to the nearest second) the slowest 5% complete their shopping in, what would
that time be?
2) The scores of a reference population on the Wechsler Intelligence Scale for Children (WISC) are normally
distributed with
A school district classified children as “gifted” if the WISC score exceeds
135. There are about 1300 sixth-graders in the school district. About how many are gifted?
RV:
1) Hens usually begin laying eggs when they are about 6 months old. Young hens tend to lay smaller eggs,
often weighing less than the desired minimum weight of 54 grams.
a) The average weight of the eggs produced by the young hens is 50.9 grams, and only 28% of their eggs
exceed the desired minimum weight. If a Normal model is appropriate, what would the standard deviation of
the egg weights be?
b) By the time these hens have reached the age of 1 year, the eggs they produce average 67.1 grams, and
98% of them are above the minimum weight. What is the standard deviation for the appropriate Normal model
for these older hens?
c) Are egg sizes more consistent for the younger hens or the older ones? Explain.
2) The weight of adult male grizzly bears living in the wild in the continental United States is approximately
normally distributed with a mean of 500 pounds and a standard deviation of 50 pounds. The weight of adult
female grizzly bears is approximately normally distributed with a mean of 300 pounds and a standard deviation
of 40 pounds. Approximately, what would be the weight of a female grizzly bear with the same standardized
scores (z-score) as a male grizzly bear with a weight of 530 pounds (2012)?
a) 276 pounds
b) 324 pounds
c) 330 pounds
d) 340 pounds
e) 530 pounds