All exercises today will be real-life applications of finding probabilities
for normal distributions. Here are tips:
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All probabilities will be if x will fall within a given interval.
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Draw the normal curve and shade the correct area.
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z-scores are multipliers, not probabilities. Avoid writing that z
equals a number a probability.
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Take note which measurements come up as normal distributed
random variables.
Notes
Areas are not changed with a transformation to a standard normal
curve.
Example 1
Notes
Find the probability that the member selected is from the shaded area
of the graph. Randomly select an SAT writing score. Assume x is
normally distributed.
µ = 493, σ = 111 Find P(200 < x < 450).
Example 2
Find the probability that the member selected is from the shaded area
of the graph. Randomly select an SAT math score. Assume x is
normally distributed.
µ = 515, σ = 116 Find P(670 < x < 800).
Notes
Example 3
Notes
Find the probability that the member selected is from the shaded area
of the graph. Randomly select a U.S. man’s total cholesterol. Assume
x is normally distributed.
µ = 209, σ = 37.8 Find P(220 < x < 255).
Example 4
Notes
Find the probability that the member selected is from the shaded area
of the graph. Randomly select a U.S. woman’s total cholesterol.
Assume x is normally distributed.
µ = 197, σ = 37.7 Find P(190 < x < 215).
Example 5
A survey indicates that people use their mobile phones an average of
1.5 years before buying an upgrade. The standard deviation is 0.25
years. A mobile phone user is selected at random. Find the
probability that the user will use their phone for less than 1 year
before buying a new one.
Notes
Example 6
Notes
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 lengths of time are normally distributed and
represented by the variable x. A shopper enters the store.
What is the probability that the shopper will be in the store between
24 and 54 minutes?
What is the probability that the shopper will be in the store more
than 39 minutes?
Interpret the answers if 200 shoppers enter the store. How many
shoppers would you expect to be in the store for each interval of time?
Example 7
Notes
Triglycerides are a type of fat in the bloodstream. The mean
triglyceride level in the U.S. is 134 milligrams per deciliter. Assume
the triglyceride levels of the population of the U.S. are normally
distributed, with a standard deviation of 35 milligrams per deciliter.
You randomly select a person from the U.S. What is the probability
that the person’s triglyceride level is less than 80?
Assignment
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§5.2: 11, 12, 16, 18, 21, 22, 26
Notes