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Cards (198)
What type of probability distribution was studied in the previous chapter?
Discrete
probability distribution
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Why can't all possible outcomes of continuous variables be listed?
Because continuous variables can take an
infinite number
of
values
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How is probability represented for continuous variables?
By the area under a
curve
called the
probability density function
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What is the probability that an observation from a continuous distribution is exactly equal to a specific value?
Zero
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What are the two conditions for a curve to be used as a probability density function?
The
total area
under the curve must be
1.
The curve must not take
negative
values (must not go
below
the horizontal axis).
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What is the main characteristic of a normal distribution?
It has a
high
probability density
close
to the mean
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What are some examples of variables that likely follow a normal distribution?
Heights
of adult females, lengths of leaves, widths of car doors, times taken by 12-year-old boys to run
100
m
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What is the standard normal distribution?
A normal distribution with mean
0
and standard deviation
1
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What is the equation of the probability density function for the standard
normal distribution
?
f(z)
=
1
2
π
e
−
z
2
2
\frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}}
2
π
1
e
−
2
z
2
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What does the table of the
normal distribution
function provide?
The probability that a normally distributed random variable Z is
less
than or
equal
to z
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How do you find the probability for a positive z-value using the normal distribution table?
Locate the
row
for the integer part and the
column
for the decimal part
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What is the probability that an observation from a standard normal distribution is less than 1.36?
0.91309
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How can you estimate the probability for a z-value given to more than two decimal places?
By using
interpolation
between the
closest
values in the table
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How do you find the probability of a value greater than a specific z-value?
Use the fact that the total area under the curve is 1 and
subtract
the probability of the z-value from
1
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What is the relationship between negative z-values and positive z-values in the normal distribution?
Negative z-values can be derived from
positive
z-values due to
symmetry
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How do you find the probability that z lies between two values?
Use the
areas
to the left of the two z-values and
subtract
them
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What is the process for standardizing a normal variable?
1.
Subtract
the
mean
from the value of interest.
2.
Divide
the result by the
standard deviation.
3. This gives the
z-score.
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How do you calculate the z-score for a value of 60 cm with a mean of 50 cm and standard deviation of 5 cm?
z
=
60
−
50
5
=
\frac{60 - 50}{5} =
5
60
−
50
=
2
2
2
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What is the z-score for a value of 47 cm with a mean of 50 cm and standard deviation of 5 cm?
z
=
47
−
50
5
=
\frac{47 - 50}{5} =
5
47
−
50
=
−
0.6
-0.6
−
0.6
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What is the first step to find the probability that a bird has a wingspan less than 17 cm?
Calculate the
z-score
for
17
cm
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If the mean wingspan is 14.1 cm and the standard deviation is 1.7 cm, what is the z-score for a wingspan of 17 cm?
z
=
17
−
14.1
1.7
≈
1.71
\frac{17 - 14.1}{1.7} \approx 1.71
1.7
17
−
14.1
≈
1.71
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What is the probability of a wingspan less than 17 cm when the z-score is 1.71?
0.956
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What are the steps to find the probability for a normal distribution with given mean and standard deviation?
1.
Standardize
the value to find the
z-score.
2. Use the
z-score
to find the
corresponding
probability from the standard normal distribution table.
3.
Interpret
the probability based on the
context
of the problem.
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How do you find the probability that a randomly selected customer has a chest measurement less than 103 cm with a mean of 101 cm and standard deviation of 5 cm?
Calculate the
z-score
and then find the
corresponding
probability
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What is the z-score for a chest measurement of 98 cm with a mean of 101 cm and standard deviation of 5 cm?
z
=
98
−
101
5
=
\frac{98 - 101}{5} =
5
98
−
101
=
−
0.6
-0.6
−
0.6
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What is the probability that a customer has a chest measurement of 98 cm or more?
0.726
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How do you find the probability between two chest measurements, 95 cm and 100 cm?
Calculate the
z-scores
for both measurements and find the
area
between them
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What is the z-score for a chest measurement of 95 cm with a mean of 101 cm and standard deviation of 5 cm?
z
=
95
−
101
5
=
\frac{95 - 101}{5} =
5
95
−
101
=
−
1.2
-1.2
−
1.2
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What is the z-score for a chest measurement of 100 cm with a mean of 101 cm and standard deviation of 5 cm?
z
=
100
−
101
5
=
\frac{100 - 101}{5} =
5
100
−
101
=
−
0.2
-0.2
−
0.2
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What is the probability between 90 cm and 110 cm for a chest measurement with mean 101 cm and standard deviation 5 cm?
0.950
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How do you ensure accuracy when finding probabilities from tables?
Keep as many
figures
as possible during
calculations
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What should you do if a question asks for a probability less than a specific value?
Use the value given without
rounding
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What are the key steps to solve problems involving normal distributions?
1. Identify the
mean
and
standard deviation.
2. Standardize the value to find the
z-score.
3. Use the
z-score
to find the corresponding probability.
4.
Interpret
the results in the
context
of the problem.
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What is the mean and standard deviation of the normal distribution for the wingspans of a population of birds?
Mean is
14.1
cm and standard deviation is
1.7
cm
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How do you find the probability that an item chosen from a normal distribution with mean 19.6 cm and standard deviation 1.9 cm is less than
20.4
cm?
Calculate the
z-score
and find the
corresponding
probability
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What is the z-score for a measurement of 20.4 cm with a mean of 19.6 cm and standard deviation of 1.9 cm?
z
=
20.4
−
19.6
1.9
≈
0.421
\frac{20.4 - 19.6}{1.9} \approx 0.421
1.9
20.4
−
19.6
≈
0.421
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How do you find the probability that an item is more than 22.0 cm in a normal distribution with mean 19.6 cm and standard deviation 1.9 cm?
Calculate the
z-score
and
subtract
the probability from 1
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What is the z-score for a measurement of 22.0 cm with a mean of 19.6 cm and standard deviation of 1.9 cm?
z
=
22.0
−
19.6
1.9
≈
1.263
\frac{22.0 - 19.6}{1.9} \approx 1.263
1.9
22.0
−
19.6
≈
1.263
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How do you find the probability that a measurement is between 19.0 and 21.0 cm?
Calculate the
z-scores
for both measurements and find the
area
between them
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What is the z-score for a measurement of 19.0 cm with a mean of 19.6 cm and standard deviation of 1.9 cm?
z
=
19.0
−
19.6
1.9
≈
−
0.316
\frac{19.0 - 19.6}{1.9} \approx -0.316
1.9
19.0
−
19.6
≈
−
0.316
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See all 198 cards
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