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standard deviation - height of basketball players













































The mean height of players in a basketball competition is 184 cm.

If the standard deviation is 5 cm, what percentage of them are likely to be:

a. taller than 189 cm

b. taller than 179 cm

c. between 174 cm and 199 cm

d. over 199 cm tall

Comments for standard deviation - height of basketball players

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Jul 22, 2014
normal curve - standard deviation
by: Staff


Answer



Part I

The following graph shows the relationship between the average () and the standard deviation (σ) using a normal curve.




Probability & Standard Deviation 

*** Click to enlarge image ***





For this problem:

= 184 cm

σ = 5 cm



When these values are added to the graph, the result is:




 Probability & Standard Deviation of basketball players’ height 

*** Click to enlarge image ***





a. What percentage of the basketball players are likely to be taller than 189 cm ?

Graphically, the percentage of basketball players taller than 189 cm is the area under the curve to the right of 189 cm.



The percentage of basketball players taller than 189 cm is the area under the curve to the right of 189 cm 

*** Click to enlarge image ***



The normal curve is symmetrical. The area under the curve from to the right is 0.5. 0.5 is the probability that a player’s height is 184 cm or more.

To determine the area from + one standard deviation to the right, use the following table of probabilities to determine the area under the curve from to one standard deviation to the right. Once you have determined what that probability is, SUBTRACT it from 0.5.


To determine the area under the curve from to specified standard deviations to the right, click the following link and use the z-table which is displayed:

Standard Normal (Z) Table from StatSoft Electronic Statistics Textbook

(For your information the same z-table can be recreated on your computer using an Excel Data Table)


Standard Normal z-table showing 1 standard deviation 

*** Click to enlarge image ***




0.5 – 0.3413 = 0.1587

Since the probability is 0.1587, the percentage probability is 15.87%.

The percentage of the basketball players which are likely to be taller than 189 cm is 15.87%.



You can also use Excel to calculate this value:

1. Open an Excel spread sheet
2. Click on an empty” cell
3. Set the formatting of the cell to 4 decimal places
4. enter “= 1.0 - ”
5. click the “insert tab”
6. click “insert function”
7. Type “Normdist” into the search box and then click “Go.”
8. Select “NORMDIST” from the list and then click “OK” to open the Function Arguments window.
9. Enter “189” for x
10. Enter “184” for mean
11. Enter “5” for standard deviation
12. Enter “true” for cumulative
13. the function which will appear in the formula bar at the top of the worksheet will look like this:

=1-NORMDIST(189,184,5,TRUE)

14. click OK or press C/R
15. the answer will be 0.1587 (this is the same answer we computed manually, using the tables)




------------------------------------------------------

Jul 22, 2014
normal curve - standard deviation
by: Staff


------------------------------------------------------


Part II


b. What percentage of the basketball players are likely to be taller than 179 cm ?

Graphically, the percentage of basketball players taller than 179 cm is the area under the curve to the right of 179 cm.




The percentage of basketball players taller than 179 cm is the area under the curve to the right of 179 cm 

*** Click to enlarge image ***





This problem is similar to question a.

Step 1: determine the area under the curve from one standard deviation to the left of to . In other words, determine the area under the curve from179 cm to 184 cm.

P (- 1 σ < Z < 0 σ ) = P (- 1 σ < Z < ) = P (179 < Z < 184)

Use the same table of values which was used in question a. The area under the curve (probability) from - 1 σ to is the exactly same as the area from to + 1 σ because the normal curve is symmetrical.




Standard Normal z-table showing 1 standard deviation 

*** Click to enlarge image ***



P (179 cm < Z < 184 cm) = 0.3413


Step 2: add the results of step 1 to 0.5 to determine the area under the curve from 179 cm or more. This will be the probability that a basketball player is greater than 179 cm tall.


(The normal curve is symmetrical. The area under the curve from to the right is 0.5. 0.5 is the probability that a player’s height is 184 cm or more.)


P (- 1 σ < Z ) = P (- 1 σ < Z < ) + P ( < Z)

P (179 < Z) = P (174 < Z < 184) + P (184 < Z)

P (179 < Z) = 0.3413 + 0.5

P (179 cm < Z) = 0.8413


Since the probability is 0.8413, the percentage probability is 84.13%.

The percentage of the basketball players which are likely to be taller than 179 cm is 84.13%.


You can also use Excel to calculate this value:

You can also use Excel to calculate this value:

1. Open an Excel spread sheet
2. Click on an empty” cell
3. Set the formatting of the cell to 4 decimal places
4. enter “= 1.0 - ”
5. click the “insert tab”
6. click “insert function”
7. Type “Normdist” into the search box and then click “Go.”
8. Select “NORMDIST” from the list and then click “OK” to open the Function Arguments window.
9. Enter “179” for x
10. Enter “184” for mean
11. Enter “5” for standard deviation
12. Enter “true” for cumulative
13. the function which will appear in the formula bar at the top of the worksheet will look like this:

=1-NORMDIST(179,184,5,TRUE)

14. click OK or press C/R
15. the answer will be 0.8413 (this is the same answer we computed manually, using the tables)



c. What percentage of the basketball players are likely to be between 174 cm and 199 cm tall ?

Graphically, the percentage of basketball players between 174 cm and 199 cm tall is the area under the curve between 174 cm and 199 cm.


 The percentage of basketball players whose height is between 174 cm and 199 cm is the area under the curve between 174 and 199 cm 

*** Click to enlarge image ***




This is a three step solution, similar to question b.

Step 1: determine the area under the curve from two standard deviations to the left of to . In other words, determine the area under the curve from174 cm to 184 cm.

P (- 2 σ < Z < 0 σ) = P (- 2 σ < Z < ) = P (174 < Z < 184)



------------------------------------------------------

Jul 22, 2014
normal curve - standard deviation
by: Staff


------------------------------------------------------


Part III


Use the same table of values which was used in part a and part b. The area under the curve (probability) from - 2 σ to is the exactly same as the area from to + 2 σ because the normal curve is symmetrical.


Standard Normal z-table showing 2 standard deviations 

*** Click to enlarge image ***




P(174 cm < Z < 184 cm) = 0.4772

Step 2: determine the area from to three standard deviations to the right of . In other words, determine the area under the curve from184 cm to 199 cm.

P ( 0 σ < Z < +3 σ) = P ( < Z < +3 σ) = P (184 < Z < 199)

Use the same table of values which was used in Step 1.


Standard Normal z-table showing 3 standard deviations 

*** Click to enlarge image ***




P (184 cm < Z < 199 cm) = 0.4987


Step 3: add the results of step 1 and step 2 to determine the area under the curve from 174 cm to 199 cm. This will be the probability that a basketball player is between 174 cm and 199 cm tall.


P (- 2 σ < Z < +3 σ) = P (174 < Z < 199)

P (- 2 σ < Z < +3 σ) = P (- 2 σ < Z < ) + P ( < Z < +3 σ)

P (174 < Z < 199) = P (174 < Z < 184) + P (184 < Z < 199)

P (174 < Z < 199) = 0.4772 + 0.4987

P(174 cm < Z < 199 cm) = 0.9759


Since the probability is 0.9759, the percentage probability is 97.59%.

The percentage of the basketball players which are likely to be between 174 cm and 199 cm tall is 97.59%.


You can also use Excel to calculate this value:

1. Open an Excel spread sheet
2. Click on an empty cell
3. Set the formatting of the cell to 4 decimal places
4. Insert the following:

=(1-NORMDIST(174,184,5,TRUE))-(1-NORMDIST(199,184,5,TRUE))

5. press C/R
6. the answer will be 0.9759 (this is the same answer we computed manually, using the tables)



d. What percentage of the basketball players are likely to be taller than 199 cm ?


Graphically, the percentage of basketball players taller than 199 cm is the area under the curve to the right of 199 cm.




 The percentage of basketball players whose height is greater than 199 cm is the area under the curve to the right of 199 cm 

*** Click to enlarge image ***






The normal curve is symmetrical. The area under the curve from to the right is 0.5. 0.5 is the probability that a player’s height is 184 cm or more.

To determine the area from + three standard deviations to the right, use the same table of probabilities which was used in questions a. b. and c. Determine the area under the curve from to three standard deviations to the right. Once you have determined what probability is, SUBTRACT it from 0.5.


Standard Normal z-table showing 3 standard deviations 

*** Click to enlarge image ***




0.5 – 0.4987 = 0.0013

Since the probability is 0.0013, the percentage probability is 0.13%.

The percentage of the basketball players which are likely to be taller than 199 cm is 0.13%.



You can also use Excel to calculate this value:

1. Open an Excel spread sheet
2. Click on an empty cell
3. Set the formatting of the cell to 4 decimal places
4. Insert the following:

=1-NORMDIST(199,184,5,TRUE)

5. press C/R
6. the answer will be 0.0013 (this is the same answer we computed manually, using the tables)





Thanks for writing.

Staff
www.solving-math-problems.com


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