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Empirical Rule
Topic 8: Standardized Scores and Normal Distributions
CCLS
standards
Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on a single count
or measurement variable.
 Represent data with plots on the real number line (dot
plots, histograms, and box plots).
 Use the mean and standard deviation of the data set to fit
it to a normal distribution and to estimate population
percentages. Recognize that there are data sets for
which such a procedure is not appropriate. Use
calculators, spreadsheets, and tables to estimate areas
under the normal curve.
Making Inferences and Justifying Conclusions
Understand and evaluate random processes underlying
statistical experiments.
 Decide whether a specified model is consistent with
results from a given data-generating process, eg, using
simulation. For example, a model says a spinning coin falls
heads up with probability 0.5. Would a result of 5 tails in a
row cause you to question the model?
Technology http://bcs.whfreeman.com/sris/#t_730892____
Appendix C: Using Fathom
CW/HW
13, 19
During her senior year, Keri has determined that her
distribution of discus distances is roughly symmetric,
unimodal, and bell-shaped, with a mean of 113 feet and a
standard deviation of 3.1 feet.
a. Sketch what this distribution should look like be drawing
a bell-shaped curve and labeling the mean, mean ±1SD,
mean ±2SD.
b. Between which two values would you expect the middle
95% of her PERFORMANCES to be?
c. Is it likely that Keri will achieve her personal goal for the
season (120 feet)? Explain.
Empirical Rule
Name: ______________________________
Probability & Statistics
1.
Date: _____
CW/HW #31
The lifespans of turtles in a particular zoo are normally distributed.
The average turtle lives 91 years; the standard deviation is 21.4 years.
Use the empirical rule (68-95-99.7%) to estimate the probability of a
turtle living longer than 48.2 years.
2. During the 2008-2009 regular season, basketball player Kobe Bryant of
the Los Angeles Lakers averaged 26.8 points per game with a
standard deviation of 8.6 points per game. Furthermore, the
distribution of his points is roughly symmetric, unimodal, and bellshaped.
a. Sketch what this distribution should look like be drawing a bellshaped curve and labeling the mean, mean ± 1SD, and mean ±
2SD.
b. Between which two values would you expect the middle 68% of his
PERFORMANCES to be?
c. What percent of his PERFORMANCES would you expect to be
below 9.6 points? Explain.
d. Calculate and interpret the z-score for his best game: 61 points.
Would you consider this an exceptional PERFORMANCE?
Empirical Rule
e. Explain.
3.
The May 2010 edition of Consumer Reports magazine reported the
usable capacity of 36 side-by-side refrigerators. The mean capacity
was 15.825 cubic feet, with a standard deviation of 1.217 feet.
a. If the distribution of usable capacity can be modeled by a Normal
distribution, about how many refrigerators will be within 1
standard deviation of the mean?
About how many refrigerators will be within 2 standard deviations?
b. The 36 measurements of usable capacity are shown below.
Calculate the number of refrigerators with a usable capacity
within 1 standard deviation of the mean and the number within 2
standard deviations of the mean:
12.9 13.7 14.1
15.2 15.3
15.3
16.0 16.2 16.2
17.0 17.0 17.2
14.2 14.2 14.5 14.5
15.3 15.3 15.5 15.6
16.3 16.4 16.5 16.6
17.4 17.4 17.9 18.4
14.6 14.7 15.1
15.6 15.8 16.0
16.6 16.6 16.8
c. Using your answers to parts (a) and (b), what can you say about the
shape of the distribution of usable capacity?
Empirical Rule
Name: ______________________________
Probability & Statistics
Date: _____
CW/HW #32
After years of practicing at a local bowling alley, Allan has
determined that his distribution of bowling scores is roughly
symmetric, unimodal, and bell-shaped, with a mean of 182 points and a
standard deviation of 23 points.
a. Sketch what this distribution should look like by drawing a bellshaped curve and labeling the mean, mean ±1SD, and mean ±2SD.
b. Between which two values would you expect the middle 68% of his
PERFORMANCES to be?
c. If Allan recorded his last 46 games, how many games should be
within 2 standard deviations of the mean?
d. Calculate and interpret the z-score of one of Allan’s game of 113
points.
e. What percent of his PERFORMANCES would you expect to be
above 113 points?
f. How likely is it that Allan bowled 113 points?
g. How likely is it that Allan will roll a perfect game (300 points), just
Empirical Rule
by RANDOM CHANCE?