Download Day 2 Notes - Honors Math 3

U6 - DAY 2 WARM UP!
• Given the times required for a group of students to complete the
physical fitness obstacle course result in a normal curve, and that
the mean time 21 minutes and the standard deviation is 4…
Use the Empirical Rule to Answer the FOLLOWING!
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•
•
•
•
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What percent took longer than 29 minutes? 2.5%
What percent took less than 29 minutes? 97.5%
What percent took between 13 and 29 minutes? 95%
What percent took between 13 and 25 minutes? 81.5%
What percent took longer than 17 minutes? 84%
What time would a student need to have to be in the fastest 2.5%
< 13 minutes
Normal Distribution Probabilities
with our Calculators!!!
If you are looking for values that
are not exactly on the standard
deviation lines, there are two
ways to find the probability –
you can use a normal
distribution chart OR you can use
a calculator. For this course we
will ONLY use the calculator!!
Normal Distribution Probability
Example #1:
A Calculus exam is given to 500
students. The scores have a
normal distribution with a mean of
78 and a standard deviation of 5.
What percent of the students
have scores between 82 and 90?
DRAW A PICTURE!!!!!
TI 83/84 directions:
Normal Distribution Probability
Example #1 A calculus exam is given to 500 students. The scores
have a normal distribution with a mean of 78 and a
standard deviation of 5. What is the probability that a
student has a score between 82 and 90?
a. Press [2nd][VARS](DISTR) [2] (normalcdf)
b. Press [82] [,] [90] [,] [78] [,] [5] [)][Enter]
normalcdf(x1, x2, μ, σ)
normalcdf(82,90,78,5)
= .2036578048
There is a 20.37%
probability that a student
scored between 82 and 90
on the Calculus exam.
Normal Distribution Probability
Example
A calculus exam is given to 500
#1: students. The scores have a normal
distribution with a mean of 78 and a
standard deviation of 5.
How many students have scores
between 82 and 90?
Using the probability previously found:
500 * .2037 = 101.85
There are about 102 students who scored
between 82 and 90 on the Calculus exam.
Normal Distribution Probability
Ex#1:
A calculus exam is given to 500
students. The scores have a normal
distribution with a mean of 78 and a
standard deviation of 5.
What percent of the students have
scores above 60?
Hint: Use 1E99 for upper limit (2nd/EE)
normalcdf(60,1E99,78,5)
= .9998408543
~ 99.98%
Normal Distribution Probability
Example #2 A PreCalculus exam is given to 500 students.
The scores have a normal distribution with a
mean of 78 and a standard deviation of 5. How
many students have scores above 70?
TI 84
2nd/EE
Normalcdf(70,1E99,78,5)
= .9452007106
500*.9452= 472.6
About 473
students have a
score above 70 on
the Pre-Calculus
exam.
Normal Distribution Probability
Example #3:
Find the probability of scoring below a 1400
on the SAT if the scores are normally
distributed with a mean of 1500 and a
standard deviation of 200.
Hint: Use -1E99 for lower limit
Normal Distribution Probability
Example #3 Find the probability of scoring below a 1400 on
the SAT if the scores are normal distributed
with a mean of 1500 and a standard deviation of
200.
normalcdf(-1E99,1400,1500,200)
= .3085375322
There is a
30.85%
probability that
a student
will score below
a 1400 on the
SAT.
Example #4
Kenny and his English Test!
• Scores of each of the previous English tests were
normally distributed with a mean of 76 and
standard deviation of 4.5. Kenny will be taking the
test tomorrow.
– What is the probability of Kenny getting at least 70 on
the test? (He wants to pass!)
~ 90.9%
– What is the probability that he makes between a 93 and
a 100? (He really wants and A!) ~ .008%
– What is the probability that he makes less than a 50?
(Oh no…his parents are going to be angry!!) ~ .0000004%
Now let’s think Backwards!!
• Example #5 Scores of each of the previous
calculus tests were normally distributed with a
mean of 86 and standard deviation of 2.2. How
high must a student’s score be to score in the top
20%?
Draw a picture…it will help!
– 2nd /vars/ invNorm (area to the left, μ, σ)
– invNorm(0.8, 86, 2.2) = ~ 87.9 or 88
PERCENTILES
(Use invNorm with percentiles)
• A percentile is the score at which a specified
percentage of scores in a distribution fall below.
• To say a score of 53 is in the 80th percentile is
to say that 80% of all scores are less than 53
More Backwards Questions…
Ex #6 Heights are generally found to be normally distributed.
Assume that the average adult female height is 65.5 inches
with a standard deviation of 3 inches.
2nd /vars/ invNorm (area to the left, μ, σ)
– What height do 90% of women fall below? 69.3 in.
– What height do 60% of women measure above? 64.7 in.
– What height is the 75th percentile of women? 67.5 in.
– What two heights, symmetric about the mean, contain
80% of all women's heights? 61.7 in. and 69.3 in.
HOMEWORK!
Day 2 HW