Transcript 4.4-4.5 Notes

Sta220 - Statistics
Mr. Smith
Room 310
Class #13
Section 4.4-4.5
The graphical form of the probability
distribution for a continuous random variable is
a smooth curve.
Definition
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Figure 4.11 A probability f(x) for a continuous
random variable x
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One of the most common observed continuous
random variable has a bell-shaped probability
distribution (or bell curve).
Figure 4.13 A normal probability distribution
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Figure 4.14 Several normal distributions with
different means and standard deviations
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The normal distribution plays a very important
role in the science of statistical inference.
You can determine the appropriateness of the
normal approximation to an existing population
of data by comparing the relative frequency
distribution of a large sample of the data with
the normal probability distribution.
To graph the normal curve, we have to know the
numerical values of 𝜇 and 𝜎.
Computing the area over intervals under the
normal probability distribution is difficult task,
so we will use the computed areas listed in Table
III of Appendix A.
Figure 4.15 Standard normal distribution:
m = 0, s = 1
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Definition
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Table 4.5
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Example 4.15A
Find the probability that the standard normal
random variable z falls between -1.33 and +1.33.
Figure 4.16 Areas under the standard normal
curve for Example 4.15
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Solution
Example 4.15B
Find the probability that the standard normal
random variable z falls between 1.00 and 2.50.
Solution
Example 4.16
Find the probability that a standard normal
random variable exceeds 1.64;
that is, find P(z > 1.64).
Figure 4.18 Areas under the standard normal
curve for Example 4.16
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Solution
Example 4.17
Find the probability that a standard normal
random variable lies to the left of .67.
Figure 4.19 Areas under the standard normal
curve for Example 4.17
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Solution
To apply Table III to a normal random variable x
with any mean 𝜇 and any standard deviation 𝜎,
we must first convent the value of x to a z-score.
This z-score transformation allows all normal
distributions to be solved with the use of the
standard normal distribution.
Definition
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Example 18
Suppose the random variable x is best described
by a normal distribution with 𝜇 = 25 and
𝜎 = 5. Find the z-score that corresponds to
the x value:
x = 37.5.
Solution
Example 19
Find a value 𝑧0 of the standard normal random
variable z such that:
a) 𝑃 𝑧 ≤ 𝑧0 = 0.0401
b) 𝑃(−𝑧0 ≤ 𝑧 ≤ 𝑧0 ) = .90
Solution
Example 4.20
Suppose x is a normal distributed random
variable with 𝜇 = 30 and 𝜎 = 8. Find a value
𝑥0 of the random variable x such that:
a) 𝑃(𝑥 ≥ 𝑥0 ) = .5.
b) 80% of the values x are less than 𝑥0 .
Solution
STOP
Sta220 - Statistics
Mr. Smith
Room 310
Class #14
Section 4.4-4.5(Part 2)
Problem 4. 21
Assume that the length of time, x, between
charges of a cellular phone is normally
distributed with a mean of 10 hours and a
standard deviation of 1.5 hours. Find the
probability that the cell phone will last between
8 and 12 between charges.
Solution
Procedure
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Problem 4. 22
Suppose an automobile manufacturer introduces a
new model that has an advertised mean in-city
mileage of 27 miles per gallon. Although such
advertisements seldom report any measure of
variability, suppose you write the manufacturer for
details on the tests and you find that the standard
deviation is 3 miles per gallon. This information
leads you to formulate a probability model for the
random variable x, the in-city mileage for this car
model. You believe that the probability distribution
of x can be approximated by a normal distribution
with a mean of 27 and a standard deviation of 3.
a. If you were to buy this model of automobile,
what is the probability that you would
purchase one that averages less than 20
miles per gallon for in-city driving?
b. Suppose you purchase one of these new
models and it does get less than 20 miles per
gallon for in-city driving. Should you conclude
that you probability model is incorrect?
Solution
Part a.
Figure 4.21 Area under the normal curve for
Example 4.21
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Solution
Part B.
There are two possibilities that could exists:
1. The probability model is correct. You simply were
unfortunate to have purchased one of the cars in the
1% that get less than 20 miles per gallon in the city.
2. The probability model is incorrect. Perhaps the
assumption of a normal distribution is unwarranted,
or the mean of 27 is an overestimate, or the stand
deviation of 3 is an underestimate, or some
combination of these errors occurred.
Keep in mind we have no way of knowing with
certainty which possibility is correct, but the
evidence points to the second on. We are again
relying on the rare-event approach to statistical
inference that we introduced earlier.
In applying the rare-event approach, the calculated
probability must be small (say, less than or equal to
0.05) in order to infer that the observed event is,
indeed, unlikely.
Problem 4.22
Suppose that the scores x on a college entrance
examination are normally distributed with a
mean of 550 and the standard deviation of 100.
A certain prestigious university will consider for
admission only those applicants whose scores
exceed the 90th percentile of the distribution.
Find the minimum score an applicant must
achieve in order to receive consideration for
admission to the university.
Figure 4.22 Area under the normal curve for
Example 4.22
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Solution
Thus, the 90th percentile of the test score
distribution is 678. That is to say, an applicant
must score at least 678 on the entrance exam to
receive consideration for admission by the
university.