If there is one prayer that you should pray/sing every day and every hour, it is the LORD's prayer (Our FATHER in Heaven prayer)
It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it. - Samuel Dominic Chukwuemeka

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Chukwuemeka



Solved Examples: Measures of Position

Prerequisite: Introductory Statistics
Calculators:
Vertical Data Entry: Measures of Position
Horizontal Data Entry: Measures of Position

Samuel Dominic Chukwuemeka (SamDom For Peace) For ACT Students
The ACT is a timed exam...$60$ questions for $60$ minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for any wrong answer.

For SAT Students
Any question labeled SAT-C is a question that allows a calculator.
Any question labeled SAT-NC is a question that does not allow a calculator.

For JAMB Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.

For WASSCE Students
Any question labeled WASCCE is a question for the WASCCE General Mathematics
Any question labeled WASSCE-FM is a question for the WASSCE Further Mathematics/Elective Mathematics

For NSC Students
For the Questions:
Any space included in a number indicates a comma used to separate digits...separating multiples of three digits from behind.
Any comma included in a number indicates a decimal point.
For the Solutions:
Decimals are used appropriately rather than commas
Commas are used to separate digits appropriately.

Solve all questions.
Show all work.

(1.) Students A, B, C, D, and E had their SAT (Scholastic Assessment Test) scores converted to z-scores.
Their z-scores are −2.00, −1.00, 0.00, 1.00, and 2.00 respectively.
Who made the highest of the five scores?


Student $E$ made the highest test score because the score is $2.00$ standard deviations above the mean.
(2.) Two friends Agnes and Agatha were arguing on who made the better score.
Agnes made a 73% on her Chemistry test while Agatha made a 46% on her Biology test.
The Biology test scores have a mean of 65% and a standard deviation of 10%, while the Chemistry test scores have a mean of 95% and a standard deviation of 11%.
Who made the better score?


$ z = \dfrac{x - \mu}{\sigma} \\[5ex] \underline{Agnes'\:\:Chemistry\:\:Test} \\[3ex] x = 73 \\[3ex] \mu = 95 \\[3ex] \sigma = 11 \\[3ex] z = \dfrac{73 - 95}{11} \\[5ex] z = -\dfrac{22}{11} \\[5ex] z = -2.00 \\[3ex] \underline{Agatha's\:\:Biology\:\:Test} \\[3ex] x = 46 \\[3ex] \mu = 65 \\[3ex] \sigma = 10 \\[3ex] z = \dfrac{46 - 65}{10} \\[5ex] z = -\dfrac{19}{10} \\[5ex] z = -1.90 \\[3ex] -1.90 \gt -2.00 \\[3ex] $ Therefore, Agatha's Biology test score is better because it has a higher $z-score$ than Agnes' Chemistry test score.

Note the responses of your students.
If some of them said that Agnes' test score was better, re-teach the concept of $z-scores$
(3.) IQ (Intelligence Quotient) scores are measured with a test designed so that the mean is 107 and the standard deviation is 19.

(a.) What are the z-scores that separate the unusual IQ scores from the usual IQ scores?

(b.) What are the IQ scores that separate the unusual IQ scores from the usual IQ scores?


(a.) A data score is usual if $-2.00 \le z \le 2.00$
A data value is unusual if the $z-score \lt -2.00$ OR the $z-score \gt 2.00$
The lower $z-score$ boundary is $-2.00$
The upper $z-score$ boundary is $2.00$
This implies that $-2.00$ and $2.00$ are the $z-scores$ that separate the unusual $IQ$ scores from the usual $IQ$ scores

(b.) To find the $IQ$ scores (data values), we need to express them in terms of the $z-scores$.

$ z = \dfrac{x - \mu}{\sigma} \\[5ex] \rightarrow x - \mu = z\sigma \\[3ex] x = z\sigma + \mu \\[3ex] Lower\:\:IQ\:\:score\:\:boundary = -2(19) + 107 = -38 + 107 = 69 \\[3ex] Upper\:\:IQ\:\:score\:\:boundary = 2(19) + 107 = 38 + 107 = 145 \\[3ex] Usual\:\:IQ\:\:scores:\:\: 69 \le IQ \le 145 \\[3ex] Unusual\:\:IQ\:\:scores:\:\: IQ \lt 69 \:\:OR\:\: IQ \gt 145 \\[3ex] $ Therefore 69 and 145 are the IQ scores that separate the unusual IQ scores from the usual IQ scores
(4.) At one time, President Barack Obama had a net worth of 3,670,505.00
The 17 members of the Executive Branch had a mean net worth of 4,939,455.00 with a standard deviation of 7,775,948.00 (OpenSecrets.org)
(a.) What is the difference between the mean net worth of all the Executive Branch members and President Obama's net worth?
(b.) How many standard deviations is the difference?
(c.) What are the z-score of President Obama's net worth?
(d.) Is President Obama's net worth usual or unusual?


$ (a.)\:\: Difference = 4939455 - 3670505 = \$1,268,950.00 \\[3ex] (b.)\:\: x = 4939455 \\[3ex] \mu = 3670505 \\[3ex] \sigma = 7775948 \\[3ex] z = \dfrac{x - \mu}{\sigma} \\[5ex] z = \dfrac{4939455 - 3670505}{7775948} \\[5ex] z = \dfrac{1268950}{7775948} \\[5ex] z = 0.16 \\[3ex] The\:\:difference\:\:is\:\:0.16\:\:standard\:\:deviations \\[3ex] (c.)\:\: \underline{Obama's\:\:worth} \\[3ex] x = 3670505 \\[3ex] \mu = 4939455 \\[3ex] \sigma = 7775948 \\[3ex] z-score = \dfrac{3670505 - 4939455}{7775948} \\[5ex] z-score = -\dfrac{1268950}{7775948} \\[5ex] z-score = - 0.1631891057 \\[3ex] z-score \approx -0.16 \\[3ex] (d.)\:\:Because -2.00 \le -0.16 \le -2.00 \\[3ex] $ Obama's net worth is usual.
(5.) Timothy surveyed 36 adults in the City of Truth or Consequences, New Mexico.
He sorted their ages (in years) as seen in the data below:

Ages
18 20 20 22 23 25 27 30 31 32 36 37
38 40 40 41 43 44 48 49 49 55 56 58
61 63 63 65 69 70 70 70 71 76 77 80

Determine and interpret the quantiles of 65 years.


Sample size = total number of values = n = 36
Number of data values less than 65 = 27

$ Percentile\;\;of\;\;65 \\[3ex] = \dfrac{27}{36} * 100 \\[5ex] = \dfrac{2700}{36} \\[5ex] = 75th\;\;percentile \\[3ex] $ This means that 75% of the ages are lower than 65 years while 25% of the ages are higher than 65 years.
Alternatively, we can say that if the ages is divided into 100 parts; the age, 65 years separates the lowest 75% of the ages from the highest 25%.

$ Decile\;\;of\;\;65 \\[3ex] = \dfrac{27}{36} * 10 \\[5ex] = \dfrac{270}{36} \\[5ex] = 7.5th\;\;decile \\[3ex] $ If the ages dataset is divided in 10 parts, 65 years separates the lowest 7.5 of the ages from the highest 2.5.

$ Quintile\;\;of\;\;65 \\[3ex] = \dfrac{27}{36} * 5 \\[5ex] = \dfrac{135}{36} \\[5ex] = 3.75th\;\;quintile \\[3ex] $ If the ages dataset is divided in 5 parts, 65 years separates the lowest 3.75 of the ages from the highest 1.25.

$ Quartile\;\;of\;\;65 \\[3ex] = \dfrac{27}{36} * 4 \\[5ex] = \dfrac{108}{36} \\[5ex] = 3rd\;\;quartile \\[3ex] $ If the ages dataset is divided in 4 parts, 65 years separates the lowest 3 of the ages from the highest 1.
(6.) Using the Ages dataset in Question Number (5.), determine the: 75th percentile, 7.5th decile, 3.75th quintile, and the 3rd quartile.


$ n = 36 \\[3ex] xth\;\;position = \dfrac{75}{100} * 36 = \dfrac{2700}{100} = 27 \\[5ex] xth\;\;position = \dfrac{7.5}{10} * 36 = \dfrac{270}{10} = 27 \\[5ex] xth\;\;position = \dfrac{3.75}{5} * 36 = \dfrac{135}{5} = 27 \\[5ex] xth\;\;position = \dfrac{3}{4} * 36 = \dfrac{108}{4} = 27 \\[5ex] \implies \\[3ex] xth\;\;position \\[3ex] = \dfrac{27th\;\;position + 28th\;\;position}{2} \\[5ex] = \dfrac{63 + 65}{2} \\[5ex] = \dfrac{128}{2} \\[5ex] = 64 \\[3ex] $ 75th percentile = 7.5th decile = 3.75th quintile = 3rd quartile = 64

Student: Mr. C, are we not supposed to get 65 as the answer?
Teacher: Yes. But we got 64. This is the correct answer.
I want you to see it this way: 64 is the boundary for the 3rd quartile (75th percentile)
65 is included in that part. But 64 is the boundary.
Be it as it may: please
NOTE: Some textbooks/online resources calculate these formulas (formulas that deal with converting a quantile to a data value) differently.
They may have it this way:

$ xth\:\:position = \dfrac{yth\:\:Percentile}{100} * (total\:\:number\:\:of\:\:values + 1) \\[5ex] $ If we use this formula, we will get the answers we got in Question Number (5.) (which we makes sense).
However, if we decide to calculate the median of that data with this formula, it will be different from the median value calculated with the formulas in the link above (main formulas).
There is still some confusion that the main formulas should only apply to the median.
Be it as it may, please use the main formulas (which is provided for you on the tests/exams)
(7.)


(8.) ACT For a given set of data, the standard score, z, corresponding to the raw score, x, is given by $z = \dfrac{x - \mu}{\sigma}$, where μ is the mean of the set and σ is the standard deviation.
If, for a set of scores, μ = 78 and σ = 6, which of the following is the raw score, x, corresponding to z = 2?

$ F.\;\; 90 \\[3ex] G.\;\; 84 \\[3ex] H.\;\; 80 \\[3ex] J.\;\; 76 \\[3ex] K.\;\; 66 \\[3ex] $

$ \mu = 78 \\[3ex] \sigma = 6 \\[3ex] z = 2 \\[3ex] x = ? \\[3ex] z = \dfrac{x - \mu}{\sigma} \\[5ex] x - \mu = z * \sigma \\[3ex] x = z * \sigma + \mu \\[3ex] x = 2 * 6 + 78 \\[3ex] x = 12 + 78 \\[3ex] x = 90 $
(9.)



(10.) Determine the five-number summary for the dataset $$ 70\hspace{4em}71\hspace{4em}71\hspace{4em}71\hspace{4em}72 \\[3ex] 72\hspace{4em}74\hspace{4em}74\hspace{4em}76\hspace{4em}77 \\[3ex] 78\hspace{4em}78\hspace{4em}79\hspace{4em}79\hspace{4em}79 \\[3ex] $$

The data is already arranged in ascending order.

$ n = 15 \\[3ex] Min = 70 \\[3ex] xth\;\;position = \dfrac{1}{4} * 15 = 3.75 \approx 4th\;\;position = 71 \\[5ex] Q_1 = 71 \\[3ex] xth\;\;position = \dfrac{2}{4} * 15 = 7.5 \approx 8th\;\;position = 74 \\[5ex] Q_2 = 74 \\[3ex] xth\;\;position = \dfrac{3}{4} * 15 = 11.25 \approx 12th\;\;position = 78 \\[5ex] Q_3 = 78 \\[3ex] Max = 79 \\[3ex] $ The five-number summary: 70, 71, 74, 78, 79
(11.)


(12.)


(13.) Determine the five-number summary for the dataset $$ 5\hspace{3.7em}6\hspace{3.7em}6\hspace{3.3em}7\hspace{3em}10\hspace{3em}11 \\[3ex] 11\hspace{3em}12\hspace{3em}12\hspace{3em}17\hspace{3em}24\hspace{3em}29 \\[3ex] 32\hspace{3em}37\hspace{3em}42\hspace{3em}45\hspace{3em}54\hspace{3em}64 \\[3ex] $$

The data is in ascending order.

$ n = 18 \\[3ex] Min = 5 \\[3ex] xth\;\;position = \dfrac{1}{4} * 18 = 4.5 \approx 5th\;\;position = 10 \\[5ex] Q_1 = 10 \\[3ex] xth\;\;position = \dfrac{2}{4} * 18 = 9 = \dfrac{9th\;\;position + 10th\;\;position}{2} = \dfrac{12 + 17}{2} = 14.5 \\[5ex] Q_2 = 14.5 \\[3ex] xth\;\;position = \dfrac{3}{4} * 18 = 13.5 \approx 14th\;\;position = 37 \\[5ex] Q_3 = 37 \\[3ex] Max = 64 \\[3ex] $ The five-number summary is: 5, 10, 14.5, 37, 64
(14.)

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(19.) HSC Mathematics Standard 1 The heights, in centimetres, of 10 players on a basketball team are shown.

170, 180, 185, 188, 192, 193, 193, 194, 196, 202

Is the height of the shortest player on the team considered an outlier?
Justify your answer with calculations.


Fences: Lower Fence and Upper Fence determine outliers of data.
A data value is an outlier if it is less than the lower fence or greater than the upper fence.
Because the question asked for the height of the shortest player, we shall be concerned with only the Lower Fence.
Data is already sorted

$ n = 10 \\[3ex] \dfrac{1}{4} * 10 \\[5ex] = 2.5th \approx 3rd\;\;position \\[3ex] Q_1 = 185 \\[5ex] \dfrac{3}{4} * 10 \\[5ex] = 7.5th \approx 8th\;\;position \\[3ex] Q_3 = 194 \\[5ex] IQR = Q_3 - Q_1 \\[3ex] IQR = 194 - 185 \\[3ex] IQR = 9 \\[3ex] LF = Q_1 - 1.5(IQR) \\[3ex] LF = 185 - 1.5(9) \\[3ex] LF = 185 - 13.5 \\[3ex] LF = 171.5 \\[3ex] $ Because the height of the shortest player is 170 cm and 170 < 171.5; the height of the shortest player is an outlier.
(20.) WASSCE In a community of 500 people, the 75th percentile age is 65 years while the 25th percentile age is 15 years.
How many of the people are between 15 and 65 years?


$ n = 500 \\[1em] \underline{65\:years} \\[1em] 75th\:\:percentile = \dfrac{75}{100} * 500 = 75 * 5 = 375 \\[2em] \underline{15\:years} \\[1em] 25th\:\:percentile = \dfrac{25}{100} * 500 = 25 * 5 = 125 \\[2em] \underline{Between\:15\:years\:\:and\:\:65\:years} \\[1em] 375 - 125 = 250\:people $




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