Download 3-6-normal-day-2

IB Math SL Year 2
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3-6: Normal Distribution Day 2
Learning Goal: What is normal distribution and how do we answer questions that use the normal curve? What does a z-score
measure? How do you compute a z-score? How can we use what we know to determining the mean and standard deviation for
approximately normal data?
1. At a college the scores on the chemistry final exam are approximately normal distributed, with a mean of 75 and a
standard deviation of 12. The scores on the calculus final are also approximately normally distributed, with a mean
of 80 and a standard deviation of 8. A student score 81 on the chemistry final and 84 on the calculus final. Relative
to the students in each respective class, in which subject did this student do better?
What is a z-score:
A z-score is a measure of “how many standard deviations a data value is from the mean”.
It is also called a “standardized score”
Formula:
A negative z-score:
A positive z-score:
A z-score of zero:
The further from zero…
IB Math SL Year 2
Let’s Try It…
1. Marc scores 700 on the math section of the SAT exam. The distribution of SAT scores has a mean of 500 and a
standard deviation of 100.
a) What is Marc’s z-score?
b) If a person had a z-score of -1.2, what would that mean?
c) Marc scores 700 on the math section of the SAT exam. The distribution of SAT scores has a mean of 500 and a
standard deviation of 100. Alice takes the ACT math exam and scores 31 on the math portion. ACT scores have
a mean of 18 and a standard deviation of 6. Relative to their peers, who earned a higher score?
Flashback! Recall, the Normal Model that we looked at yesterday…
The Standard Normal Curve: 𝑍~N(0, 1).
The standard normal distribution is the normal distribution where 𝜇 = 0 and 𝜎 = 1. The random variable is called 𝑍. It
uses ‘𝑧’ values to describe the number of standard deviations any value is away from the mean.
The Empirical Rule:
Approximately ______% of the distribution lies within 1 standard deviation of the mean
Approximately ______% of the distribution lies within 2 standard deviation of the mean
Approximately ______% of the distribution lies within 3 standard deviation of the mean
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What percent of a standard
Normal model is found in...?
Sketch of model
a) 𝑧 < −1
b) 𝑧 > 2
c)−2 < 𝑧 < 1
d) 𝑧 < −1 𝑜𝑟 𝑧 > 2
2. Find the area under the standard normal curve:
a. Between 1 and 2 standard deviations from the mean.
b. Between 0.5 and 1.5 standard deviations from the mean.
Proportion/Percent
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3. Given that Z~N(0,1) use the GDC to find,
a. P(Z<0.65)
b. (Z>0.72)
4. Find the Z-score corresponding to the given value of X with X~N(20,9), x = 15
Finding mean and standard deviation
Sacks of potatoes with mean weight 5kg are packed by an automatic loader. In a test it was found that 10% of bags
were over 5.2 kg. Use this information to find the standard deviation of the process.
What are we given?
What are we asked to find?
Draw a picture
How can we solve this?
Let’s Try It…
5. X~N(𝜇,42) and P(X<20.5) = 0.9. Find the value of𝜇.
IB Math SL Year 2
6. X~N(𝜇, 𝜎) and P(X>58.39) = 0.0217 and P(X<41.82) = 0.0287. Find the value of𝜇. and 𝜎.
3-6 Practice
7. Given that Z~N(0,1), use the GDC to find
a. P(-1.3<Z < -0.3)
2. Given𝑍~𝑁(0,1), determine the following probabilities
a) 𝑃(𝑍 > 0.72)
b) 𝑃(𝑍 ≤ 1.8)
3. A random variable X is normally distributed with unknown mean and standard deviation, such that P(X<89) = 0.90
and P(X<94) = 0.95. Find 𝜎 and 𝜇.
IB Math SL Year 2
4. A manufacturer does not know the mean and standard deviation of the diameters of ball bearings she is producing.
However a sieving systems rejects all ball bearings larger than 2.4cm and those under 1.8cm in diameter. It is found
the 8% of the ball bearings are rejected as too small and 5.5% as too big. What I the mean and standard deviation of
the ball bearings produced?
5. The standard deviation of masses of loaves of bread is 20g. Only 1% of loaves weight less thatn 500g. Find the mean
mass of the loaves.
6. The masses of cauliflowers are normally distributed with mean 0.85kg. 74% of the cauliflouwers have mass less than
1.1kg. Find:
a) The standard deviation of cauliflowers’ masses
b) The percentage of cauliflowers with mass greater than 1kg.
IB Math SL Year 2
7. The lengths of nails are normally distributed with mean 𝜇 and standard deviation 𝜎 = 7cm. If 2.5 % of the nails
measured more than 68cm, find the value of𝜇.
8. Find the Z-score corresponding to the given value of X.
(a) X~N(38,72), x = 45
(b) X~N(162,25), x = 160
9. A roll of wrapping paper is sold as “3m long.” It is found that actually only 35% of rolls are over 3m long and that the
average length of the rolls of the wrapping paper is 2.9m. Find the value of the standard deviation of the lengths of
rolls of wrapping paper, assuming that the lenths of rolls follow a normal distruction.
10. The battery life of a certain brand of laptop batteries follow s a normal distribution with mean 16 hours and
standard deviation of 5 hours. A particular battery has a life of 10.2 hours.
(a) How many standard deviations below the mean is this battery life?
(b) What is the probability that a randomly chose battery has a life short than this?