CABT SHS Statistics & Probability - The z-scores and Problems involving Normal Distributions

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CABT Statistics & Probability – Grade 11 Lecture Presentation

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The Normal Distribution and Standard Scores

A Grade 11Statistics & Probability

Lecture

3


Normally Distributed Random Variables


A normally distributed random variable X (or X has a normal distribution) has a continuous, symmetric, bell-shaped distribution symmetric about the mean of X.

If X is normally-distributed with mean and standard deviation , we write

X ~ N(, )

The distance of the values of X from the mean is expressed in terms of the standard deviation .

Properties of the Normal Distribution1. The distribution curve

is bell-shaped.2. The curve is symmetric

about its center, the mean.

3. The mean, the median, and the mode coincide at the center.

4. The width of the curve is determined by the standard deviation of the distribution.



Properties of the Normal Distribution5. The tails of the curve

flatten out indefinitely along the horizontal axis, always approaching the axis but never touching it. That is, the curve is asymptotic to the base line.

6. The area under the curve is 1. Thus, it represents the probability or proportion or the percentage associated with specific sets of measurement values.

asymptotic to the x-axis


The Distribution of Area Under the Normal Curve- a.k.a. the empirical rule or the

“68% - 95% - 99.7%” ruleThe area under the part of a normal curve that lies:• within 1 standard deviation of the

mean is approximately 0.68, or 68%;• within 2 standard deviations, about

0.95, or 95%within 3 standard deviations, about 0.997, or 99.7%CABT Statistics & Probability – Grade 11 Lecture Presentation


The Distribution of Area Under the Normal Curve



The Standard Score or z-Scores


The STANDARD NORMAL random variable Z has a normal distribution with mean = 0 and standard deviation = 1. If Z is normally-distributed with mean 0 and standard deviation 1, we write

Z ~ N(0, 1)




QUESTION:

What comes to mind when you hear the words

STANDARD and STANDARDIZED?


What is STANDARD?


STANDARD n. a level of quality or attainment (high standard of service); an idea or thing used as a measure, norm, or model in comparative evaluations (the wages are low by today's standards)STANDARD adj. used or accepted as normal or average (standard score)

The Standard Score or z-Scores


The standard score or z-score is a measure of relative standing. It represents the distance between a given measurement X and the mean, expressed in standard deviations.


Converting Normally-Distributed Scores to z-Scores


If the distribution of a random variable X is normal with mean and standard deviation , the values x of X can be STANDARDIZED or can be converted to z-scores using this formula: xz

This formula transforms the values of the variable x into standard units or z values.





Relationship between x and z:For any population, the mean and the standard deviation are fixed. Thus, the z formula matches the z-values one-to-one with the x values (raw scores). That is, for every x value there corresponds a z-value and for each z-value there is exactly one x value.

z xx z

xz



Note on notation: If we’re talking about an entire POPULATION that is normally distributed, we use for the mean and for the standard deviation.If we’re talking about a SAMPLE of a population that is normally distributed, we use for the mean and s for standard deviation.

x


x

The z-ScoresConverting Normally-Distributed Scores to z-Scores


Note on notation: Z-score for

a populationZ-score fora sample

xz

x xz

sx



Why standardize scores?


The major purpose of standard scores is to place scores for any individual on any variable having any mean and standard deviation on the same standard scale so that comparisons can be made. Without some standard scale, comparisons across individuals and/or across variables would be difficult to make. (http://faculty.virginia.edu/PullenLab/WJIIIDRBModule/WJIIIDRBModule7.html)



Application: the NCAE


The scores in the NCAE are reported in Standard Scores and Percentile Ranks.Standard Score – where the mean is 500 and the standard deviation is 100. The highest scores are in the 700’s; the lowest scores are in the 300’s.http://www.teacherph.com/national-career-assessment-examination-ncae-overview/





http://www.teacherph.com/national-career-assessment-examination-ncae-overview/

Percentile Rank – shows the test taker’s position among all the examinees. If an examinee scores at percentile rank 99+, it means that he scored above the other 99 percent of the examinees.





http://www.teacherph.com/national-career-assessment-examination-ncae-overview/

The normal curve on the right represents a sample plot of a percentile rank (PR) in the NCAE.



1Given a normally distributed population with mean 75 and standard deviation 4, find the corresponding standard score of the following:a. 69 b. 85


69, 75, 4xxz

69 75

4 1.5z

85, 75, 4xxz

85 75

42.5z



2In a given normal distribution, the sample mean is 20.5 and the sample standard deviation is 5.4. Find the corresponding standard score of the following:a. 18.7 b. 21.3


18.7, 20.5, 5.4x x s

x xz

s

18.7 20.5

5.4 0.33z

21.3, 20.5, 5.4x x s

x xz

s

21.3 20.5

5.40.15z



3The average score in a Statistics and Probability Test is 80 with standard deviation 10. What is the standard score of the following students?JM – 97 Eljhie – 86


1 97, 80, 10x

11

xz

97 8010

1 1.7z

2 86, 80, 10x

22

xz

86 8010

2 0.6z



4A sample of bags of potato chips in a factory has an average net weight of 24.7 grams with standard deviation of 0.35 grams.What is the standard score for two samples A and B with the following weights? A – 22.6 grams B – 25.9 grams


22.6, 24.7, 1.02Ax x s

AA

x xzs

22.6 24.71.02

2.06Az

25.9, 24.7, 1.02Bx x s

BB

x xzs

25.9 24.71.02

1.18Bz



5Scores in the Precalculus Quarter Exam has mean 76 and standard deviation 4, while the Gen. Math exam has mean 75 and standard deviation 5.


If Yayie scored 90 in Precalculus and 91 in Gen. Math, in which subject is her standing better? Assume that the scores in both exams are normally distributed.



5Scores in the Precalculus Quarter Exam has mean 76 and standard deviation 4, while the Gen. Math exam has mean 75 and standard deviation 5. If Yayie scored 90 in Precalculus and 91 in Gen. Math, in which subject is her standing better? Assume that the scores in both exams are normally distributed.


90, 76, 4x

xz

90 76

43.5z

For Precalculus:

For Gen. Math 91, 75, 5x

xz

91 75

53.2z

Yayie has a better standing in Precalculus than in Gen. Math.



If the standard score z is given, the original or raw score x can be obtained by solving xz

for x, yielding the equation


x zFor a sampling distribution,

x x zs



6Find the raw score of a standard score of z = 1.2 in a normally-distributed population with mean 30 and standard deviation 4.


1.2, 30, 4z x z 30 1.2 4

25.2x

Do you have any QUESTIONs?

Probabilities and Standardized Scores



Recall that areas under normal curves correspond to probabilities or percent of scores.PROBABILITY CORRESPONDING AREA

P(X > a) to the right of aP(X < a) to the left of a

P( a < X < b) between a and bNOTE: The area won’t change even if “>” and “<” are replaced by “” and “”, respectively.




To determine the probabilities or percent that values of X in a normally-distributed population fall on a particular interval:STEP 1 – Convert the x scores to z-scores. STEP 2 – Draw the region defined by the z scores.STEP 3 – Locate the areas corresponding to the z scores using the table.STEP 4 – Find the area of the indicated region.




7In the Oral Comm test given by Sir Aldous, the mean score is 60 with standard deviation 6.

Assuming that the scores are normally-distributed, what is the probability that a randomly-selected student has a scorea. between 60 and 65? P(60 < X <

65)b. greater than 65? P(X > 65)c. between 50 and 65? P(50 < X < 65)




7a. Convert x = 60 and x = 65 to z-

scores 1 60, 60, 6x

11

xz

60 606

1 0z

2 65, 60, 6x

22

xz

65 606

2 0.83z

0.83

The area can be directly read in the table: 0.2967. Hence,P(0 < X < 65) = 0.2967




7b. Convert x = 65 to z-score

65, 60, 6x

xz

65 60

6 0.83The area corresponding to z = 0.83 is 0.2967. P(X < 65) = P(Z < 0.83)

= 0.5 + 0.2967= 0.79670.83




8In a university, the average number of years a person takes to complete a master’s degree program is 3 and the standard deviation is 4 months.

Assume the variable is normally distributed. If an individual enrolls in the program, find the probability that it will takea. more than 4 years to complete the program.b. less than 3 years to complete the program.c. between 3.8 and 4.5 years to complete the programd. between 2.5 and 3.1 years to complete the programs.




9The mean age of a population of 10,000 is 56 years old, with standard deviation of 5 years. If the ages are normally-distributed, how manya. have ages between 40 and 45 years?

b. are senior citizens?c. are teenagers?d. are ages 56 years old and below?

Standardized Scores and Percentiles



Recall that• a percentile is a measure of relative

standing or position. It is a descriptive measure of the relationship of a measurement to the rest of the data.

• a score x is in the kth percentile rank (Pk) if k% of the scores are equal or below x.




Percentile and z-scores A probability value corresponds to an area

under the normal curve. In the Table of Areas Under the Normal

Curve, the numbers in the extreme left and across the top are z-scores, which are the distances along the horizontal scale. The numbers in the body of the table are areas or probabilities.

The z-scores to the left of the mean are negative values.




Percentile and z-scores Recall that a probability corresponds

to a percent or proportion; e.g. a probability of 0.4922 is the same as a probability of 49.22%

Since x = Pk represents scores LESS that or equal to x, the region representing a percentile rank in a normal distribution is the same as P(X < x).



10In an examination, the scores are normally distributed with mean 20 and the standard deviation 2. Determine the percentile ranks of the following scores:a. 18 (15.87% - 16th

percentile)b. 25 (99.38% - 99th

percentile)


Do you have any QUESTIONs?

Summing it up!

Thank you!

CABT SHS Statistics & Probability - The z-scores and Problems involving Normal Distributions

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Transcript of CABT SHS Statistics & Probability - The z-scores and Problems involving Normal Distributions