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“Blood Pressure of Males and Females.” StatCruch, 2013. Available online at http://www.statcrunch.com/5.0/viewreport.php?reportid=11960 (accessed May 14, 2013).
“The Use of Epidemiological Tools in Conflict-affected populations: Open-access educational resources for policy-makers: Calculation of z-scores.” London School of Hygiene and Tropical Medicine, 2009. Available online at http://conflict.lshtm.ac.uk/page_125.htm (accessed May 14, 2013).
“2012 College-Bound Seniors Total Group Profile Report.” CollegeBoard, 2012. Available online at http://media.collegeboard.com/digitalServices/pdf/research/TotalGroup-2012.pdf (accessed May 14, 2013).
“Digest of Education Statistics: ACT score average and standard deviations by sex and race/ethnicity and percentage of ACT test takers, by selected composite score ranges and planned fields of study: Selected years, 1995 through 2009.” National Center for Education Statistics. Available online at http://nces.ed.gov/programs/digest/d09/tables/dt09_147.asp (accessed May 14, 2013).
Data from the San Jose Mercury News .
Data from The World Almanac and Book of Facts .
“List of stadiums by capacity.” Wikipedia. Available online at https://en.wikipedia.org/wiki/List_of_stadiums_by_capacity (accessed May 14, 2013).
Data from the National Basketball Association. Available online at www.nba.com (accessed May 14, 2013).
A z -score is a standardized value. Its distribution is the standard normal, Z ~ N (0, 1). The mean of the z -scores is zero and the standard deviation is one. If z is the z -score for a value x from the normal distribution N ( µ , σ ) then z tells you how many standard deviations x is above (greater than) or below (less than) µ .
Z ~ N (0, 1)
z = a standardized value ( z -score)
mean = 0; standard deviation = 1
To find the
K
th percentile of
X when the
z -scores is known:
k =
μ + (
z )
σ
z -score: z =
Z = the random variable for z -scores
Z ~ N (0, 1)
A bottle of water contains 12.05 fluid ounces with a standard deviation of 0.01 ounces. Define the random variable X in words. X = ____________.
ounces of water in a bottle
A normal distribution has a mean of 61 and a standard deviation of 15. What is the median?
X ~ N (1, 2)
σ = _______
2
A company manufactures rubber balls. The mean diameter of a ball is 12 cm with a standard deviation of 0.2 cm. Define the random variable X in words. X = ______________.
X ~ N (–4, 1)
What is the median?
–4
X ~ N (3, 5)
σ = _______
X ~ N (–2, 1)
μ = _______
–2
What does a z -score measure?
What does standardizing a normal distribution do to the mean?
The mean becomes zero.
Is X ~ N (0, 1) a standardized normal distribution? Why or why not?
What is the z -score of x = 12, if it is two standard deviations to the right of the mean?
z = 2
What is the z -score of x = 9, if it is 1.5 standard deviations to the left of the mean?
What is the z -score of x = –2, if it is 2.78 standard deviations to the right of the mean?
z = 2.78
What is the z -score of x = 7, if it is 0.133 standard deviations to the left of the mean?
Suppose X ~ N (2, 6). What value of x has a z -score of three?
x = 20
Suppose X ~ N (8, 1). What value of x has a z -score of –2.25?
Suppose X ~ N (9, 5). What value of x has a z -score of –0.5?
x = 6.5
Suppose X ~ N (2, 3). What value of x has a z -score of –0.67?
Suppose X ~ N (4, 2). What value of x is 1.5 standard deviations to the left of the mean?
x = 1
Suppose X ~ N (4, 2). What value of x is two standard deviations to the right of the mean?
Suppose X ~ N (8, 9). What value of x is 0.67 standard deviations to the left of the mean?
x = 1.97
Suppose X ~ N (–1, 2). What is the z -score of x = 2?
Suppose X ~ N (12, 6). What is the z -score of x = 2?
z = –1.67
Suppose X ~ N (9, 3). What is the z -score of x = 9?
Suppose a normal distribution has a mean of six and a standard deviation of 1.5. What is the z -score of x = 5.5?
z ≈ –0.33
In a normal distribution, x = 5 and z = –1.25. This tells you that x = 5 is ____ standard deviations to the ____ (right or left) of the mean.
In a normal distribution, x = 3 and z = 0.67. This tells you that x = 3 is ____ standard deviations to the ____ (right or left) of the mean.
0.67, right
In a normal distribution, x = –2 and z = 6. This tells you that x = –2 is ____ standard deviations to the ____ (right or left) of the mean.
In a normal distribution, x = –5 and z = –3.14. This tells you that x = –5 is ____ standard deviations to the ____ (right or left) of the mean.
3.14, left
In a normal distribution, x = 6 and z = –1.7. This tells you that x = 6 is ____ standard deviations to the ____ (right or left) of the mean.
About what percent of x values from a normal distribution lie within one standard deviation (left and right) of the mean of that distribution?
about 68%
About what percent of the x values from a normal distribution lie within two standard deviations (left and right) of the mean of that distribution?
About what percent of x values lie between the second and third standard deviations (both sides)?
about 4%
Suppose X ~ N (15, 3). Between what x values does 68.27% of the data lie? The range of x values is centered at the mean of the distribution (i.e., 15).
Suppose X ~ N (–3, 1). Between what x values does 95.45% of the data lie? The range of x values is centered at the mean of the distribution(i.e., –3).
between –5 and –1
Suppose X ~ N (–3, 1). Between what x values does 34.14% of the data lie?
About what percent of x values lie between the mean and three standard deviations?
about 50%
About what percent of x values lie between the mean and one standard deviation?
About what percent of x values lie between the first and second standard deviations from the mean (both sides)?
about 27%
About what percent of x values lie betwween the first and third standard deviations(both sides)?
Use the following information to answer the next two exercises: The life of Sunshine CD players is normally distributed with mean of 4.1 years and a standard deviation of 1.3 years. A CD player is guaranteed for three years. We are interested in the length of time a CD player lasts.
Define the random variable X in words. X = _______________.
The lifetime of a Sunshine CD player measured in years.
X ~ _____(_____,_____)
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