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References

“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).

Chapter review

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 .

Formula review

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 = x  –  μ σ

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

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A normal distribution has a mean of 61 and a standard deviation of 15. What is the median?

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X ~ N (1, 2)

σ = _______

2

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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 = ______________.

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X ~ N (–4, 1)

What is the median?

–4

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X ~ N (3, 5)

σ = _______

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X ~ N (–2, 1)

μ = _______

–2

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What does a z -score measure?

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What does standardizing a normal distribution do to the mean?

The mean becomes zero.

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Is X ~ N (0, 1) a standardized normal distribution? Why or why not?

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What is the z -score of x = 12, if it is two standard deviations to the right of the mean?

z = 2

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What is the z -score of x = 9, if it is 1.5 standard deviations to the left of the mean?

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What is the z -score of x = –2, if it is 2.78 standard deviations to the right of the mean?

z = 2.78

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What is the z -score of x = 7, if it is 0.133 standard deviations to the left of the mean?

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Suppose X ~ N (2, 6). What value of x has a z -score of three?

x = 20

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Suppose X ~ N (8, 1). What value of x has a z -score of –2.25?

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Suppose X ~ N (9, 5). What value of x has a z -score of –0.5?

x = 6.5

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Suppose X ~ N (2, 3). What value of x has a z -score of –0.67?

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Suppose X ~ N (4, 2). What value of x is 1.5 standard deviations to the left of the mean?

x = 1

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Suppose X ~ N (4, 2). What value of x is two standard deviations to the right of the mean?

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Suppose X ~ N (8, 9). What value of x is 0.67 standard deviations to the left of the mean?

x = 1.97

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Suppose X ~ N (–1, 2). What is the z -score of x = 2?

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Suppose X ~ N (12, 6). What is the z -score of x = 2?

z = –1.67

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Suppose X ~ N (9, 3). What is the z -score of x = 9?

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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

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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.

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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

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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.

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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

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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.

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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%

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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?

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About what percent of x values lie between the second and third standard deviations (both sides)?

about 4%

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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).

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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

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Suppose X ~ N (–3, 1). Between what x values does 34.14% of the data lie?

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About what percent of x values lie between the mean and three standard deviations?

about 50%

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About what percent of x values lie between the mean and one standard deviation?

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About what percent of x values lie between the first and second standard deviations from the mean (both sides)?

about 27%

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About what percent of x values lie betwween the first and third standard deviations(both sides)?

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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.

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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