Ch 1 Student Review 2009 - Oocities Ans
The standard normal distribution is a normal distribution of standardized values chosen z-scores. A z-score is measured in units of the standard deviation. For case, if the hateful of a normal distribution is five and the standard deviation is ii, the value 11 is three standard deviations in a higher place (or to the correct of) the mean. The calculation is as follows:
x = μ + (z)(σ) = 5 + (iii)(two) = eleven
The z-score is three.
The hateful for the standard normal distribution is zero, and the standard deviation is one. The transformation [latex]\displaystyle{z}=\frac{{ten - \mu}}{{\sigma}}[/latex] produces the distribution Z ~ N(0, 1). The value x comes from a normal distribution with hateful μ and standard deviation σ.
The following ii videos give a description of what it means to take a data set that is "normally" distributed.
Z-Scores
If X is a normally distributed random variable and X ~ North(μ, σ), then the z-score is:
[latex]\displaystyle{z}=\frac{{x - \mu}}{{\sigma}}[/latex]
The z-score tells you how many standard deviations the value x is above (to the correct of) or below (to the left of) the mean, μ.
- Values of x that are larger than the mean have positive z-scores.
- Values of ten that are smaller than the mean have negative z-scores.
- If ten equals the mean, then x has a z-score of cypher.
Note:
- The z-scores for µ+1σ and µ–1σ are +ane and –1, respectively.
- The z-scores for µ+2σ and µ–2σ are +2 and –2, respectively.
- The z-scores for µ+3σ and µ–threeσ are +iii and –3 respectively.
Example 1
Suppose 10 ~ Due north(5, six).
This says that ten is a normally distributed random variable with hateful μ = 5 and standard divergence σ = six.
1. Suppose there is a raw information,10 of 17. What is the z-score?
2. What is the z-score for raw data x = 1?
Detect that: 5 + (–0.67)(half dozen) is approximately equal to ane. (This has the blueprint μ + zσ = raw data.)
Summarizing,
- When z is positive, x is greater than or to the right of μ.
(when x is greater than μ, the corresponding z-score is positive,. ) - When zis negative, x is less than or to the left ofμ.
(When 10 is less than μ, the respective z-score is negative. )
Effort It
What is the z-score of x, when 10 = 1 andX ~ N(12,3)?
Testify Respond
[latex]\displaystyle {z}=\frac{{1-12}}{{3}} = -{iii.67} [/latex]
Case 2
Some doctors believe that a person can lose five pounds, on the boilerplate, in a month by reducing his or her fat intake and past exercising consistently. Suppose weight loss has a normal distribution. Allow X = the corporeality of weight lost(in pounds) by a person in a month. Use a standard deviation of 2 pounds. X ~ N(5, ii). Fill in the blanks.
- Suppose a person lost ten pounds in a calendar month. The z-score when 10 = 10 pounds is z = 2.five (verify). This z-score tells you thatx = 10 is ________ standard deviations to the ________ (right or left) of the hateful _____ (What is the mean?).
Bear witness Respond
This z-score tells you that data 10 is two.v standard deviations to the right of the mean five .
- Suppose a person gained three pounds (a negative weight loss). Then z = __________. This z-score tells you that datax = –iii is ________ standard deviations to the __________ (right or left) of the mean.
Prove Answer
z= –4 . This z-score tells you that information 10 = –3 is 4 standard deviations to the left of the mean.
Suppose the random variables X and Y take the following normal distributions: Ten ~ North(5, 6) and Y ~ N(2, 1).
If x = 17, then z = 2. (This was previously shown.)
If y = iv, what is z?
[latex]\displaystyle {z}=\frac{{y - \mu}}{{\sigma}} = \frac{{four-2}}{{1}}[/latex].
The z-score for y = 4 is 2.
This ways that raw data 4 is 2 standard deviations to the right of the mean.
Therefore, 10 = 17 and y = four are both ii (of their own) standard deviations to the correct of their corresponding means.
The z-score allows united states of america to compare data that are scaled differently.
To sympathise the concept, suppose 10 ~ Northward(five, six) represents weight gains for one grouping of people who are trying to gain weight in a six week period and Y ~ North(2, 1) measures the same weight proceeds for a second group of people. (A negative weight gain would be a weight loss. )
Since 10 = 17 and y= 4 are each two standard deviations to the right of their means, they represent the same, standardized weight gain relative to their means.
Try It
Fill in the blanks.
Jerome averages 16 points a game with a standard deviation of iv points. 10 ~Northward(16,four). Suppose Jerome scores x points in a game. The z–score when x = x is –1.5. This score tells you lot that 10 = 10 is _____ standard deviations to the ______(correct or left) of the mean______(What is the hateful?).
Evidence Answer
1.5, left, 16
The Empirical Dominion
If X is a random variable and has a normal distribution with mean µ and standard deviation σ,
then the Empirical Rule says the following:
- About 68% of the x values prevarication between the range betwixtµ – σ and µ + σ (inside one standard deviation of the hateful).
- About 95% of the ten values lie between the range betweenµ – 2σ and µ + 2σ (within 2 standard deviations of the mean).
- Virtually 99.vii% of the 10 values prevarication between the range betweenµ – 3σ and µ + 3σ(within iii standard deviations of the mean).
Notice that almost all thex-values/data lie within three standard deviations of the hateful.
The empirical rule is also known as the 68-95-99.7 dominion.
Example 3
The mean height of 15 to 18-year-old males from Chile from 2009 to 2010 was 170 cm with a standard difference of 6.28 cm.
Male person heights are known to follow a normal distribution.
Let X = the top of a 15 to eighteen-twelvemonth-sometime male from Chile in 2009 to 2010. Then Ten ~ N(170, half-dozen.28).
a. Suppose a 15 to 18-year-one-time male from Chile was 168 cm alpine from 2009 to 2010.
The z-score when 10 = 168 cm is z = _______.
This z-score tells you lot that 10 = 168 is ________ standard deviations to the ________ (right or left) of the mean _____.
Show Answer
a. –0.32, 0.32, left, 170
b. Suppose that the height of a 15 to 18-year-old male from Chile from 2009 to 2010 has a z-score of z = 1.27.
What is the male's height?
The z-score (z = 1.27) tells you lot that the male's top is ________ standard deviations to the __________ (right or left) of the mean.
Testify Reply
b. 177.98, 1.27, right
Try It
Apply the data in Case 3 to answer the following questions.
- Suppose a 15 to 18-year-onetime male person from Chile was 176 cm tall from 2009 to 2010. The z-score when x = 176 cm is z = _______. This z-score tells you that 10 = 176 cm is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).
- Suppose that the height of a 15 to 18-year-old male from Chile from 2009 to 2010 has a z-score of z = –2. What is the male's height? The z-score (z = –2) tells you that the male's height is ________ standard deviations to the __________ (right or left) of the mean.
Bear witness Reply
Solve the equation [latex]\displaystyle{z}=\frac{{10 - \mu}}{{\sigma}}[/latex] for 10. ten = μ + (z)(σ) for 10. x = μ + (z)(σ) z<=[latex]\displaystyle\frac{{176-170}}{{0.96}}[/latex], This z-score tells y'all that x = 176 cm is 0.96 standard deviations to the right of the mean 170 cm. X = 157.44 cm, The z-score(z = –2) tells you that the male person'southward height is two standard deviations to the left of the hateful.
Instance 4
From 1984 to 1985, the mean height of fifteen to 18-year-onetime males from Chile was 172.36 cm, and the standard deviation was half-dozen.34 cm. Let Y = the summit of 15 to 18-year-old males from 1984 to 1985. Then Y ~ N(172.36, 6.34).
The mean height of 15 to 18-twelvemonth-sometime males from Chile from 2009 to 2010 was 170 cm with a standard deviation of vi.28 cm. Male heights are known to follow a normal distribution. Let X = the pinnacle of a 15 to 18-year-old male from Chile in 2009 to 2010. And so X ~ N(170, six.28).
- Discover the z-scores for x = 160.58 cm and y = 162.85 cm.
- Translate each z-score. What tin you say well-nigh x = 160.58 cm and y = 162.85 cm?
Try It
In 2012, 1,664,479 students took the SAT exam. The distribution of scores in the verbal section of the Sat had a hatefulµ = 496 and a standard deviation σ = 114. Let 10 = a SAT exam verbal department score in 2012. Then 10 ~ Due north(496, 114).
Find the z-scores for 10ane = 325 and xii = 366.21. Interpret each z-score. What can you say well-nigh 10i = 325 and x2 = 366.21?
Bear witness Answer
The z-score for x1 = 325 is z1 = –i.14. The z-score for x2 = 366.21 is z2 = –one.14. Student 2 scored closer to the mean than Student 1 and, since they both had negative z-scores, Student two had the amend score.
Instance 5
Suppose x has a normal distribution with mean 50 and standard departure 6.
Employ empirical rule to interpret the information distribution.
Try Information technology
Suppose 10 has a normal distribution with mean 25 and standard deviation 5. Between what values of ten do 68% of the values lie?
Show Answer
Between 20 and thirty.
Example 6
From 1984 to 1985, the mean peak of 15 to xviii-yr-old males from Chile was 172.36 cm, and the standard departure was vi.34 cm.
Let Y = the meridian of xv to xviii-yr-old males in 1984 to 1985,Y ~ N(172.36, 6.34).
(The variable Y is normally distributed. )
- About 68% of the y values prevarication between what two values?
These values are ________________. The z-scores are ________________, respectively. - About 95% of the y values lie between what two values?
These values are ________________. The z-scores are ________________ respectively. - Nearly 99.7% of the y values lie between what ii values?
These values are ________________. The z-scores are ________________ respectively.
Try It
The scores on a college entrance exam have an approximate normal distribution with hateful, µ = 52 points and a standard difference, σ = 11 points.
Show Reply
Virtually 68% of the values lie betwixt the values 41 and 63.
Nigh 95% of the values lie between the values 30 and 74.
About 99.7% of the values lie between the values 19 and 85.
.
References
"Blood Pressure of Males and Females." StatCruch, 2013. Available online at http://www.statcrunch.com/5.0/viewreport.php?reportid=11960 (accessed May xiv, 2013).
"The Apply 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://disharmonize.lshtm.ac.uk/page_125.htm (accessed May xiv, 2013).
"2012 Higher-Spring Seniors Total Grouping Contour Report." CollegeBoard, 2012. Available online at http://media.collegeboard.com/digitalServices/pdf/research/TotalGroup-2012.pdf (accessed May 14, 2013).
"Digest of Educational activity Statistics: ACT score boilerplate and standard deviations by sex and race/ethnicity and per centum of ACT examination takers, by selected composite score ranges and planned fields of study: Selected years, 1995 through 2009." National Heart 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 Earth 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).
Concept 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 null and the standard difference is one. If zis the z-score for a value 10 from the normal distribution N(µ, σ) so z tells y'all how many standard deviations x is above (greater than) or beneath (less than) µ.
Formula Review
Z ~ N(0, one)
z = a standardized value (z-score)
mean = 0; standard deviation = 1
To catechumen z-score into raw information: raw data = μ + (z)σ
To convert data into z-score: [latex]\displaystyle{z}=\frac{{x - \mu}}{{\sigma}}[/latex]
Source: https://courses.lumenlearning.com/odessa-introstats1-1/chapter/the-standard-normal-distribution/
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