Question

(1 point) final scores in a mathematics course are normally distributed with a mean of 71 and a standard deviation of 13. Based on the above information and a Z-table, fill in the blanks in the table below.  

Precision and other notes: (1) Percentiles should be recorded in percentage form to three decimal places.  
(2) Note that this problem does not use the rough values of the 68-95-99.7 rule (that is, the empirical rule); instead you must use more precise Z-table values for percentiles.

test score	Z-score	Percentile
84
58
	3
		2.28

Answer 1

µ =    71          
σ =    13          
              
P( X ≤    84   ) = P( (X-µ)/σ ≤ (84-71) /13)      
=P(Z ≤   1.00   ) =   0.8413   (answer)
================

P( X ≤    58   ) = P( (X-µ)/σ ≤ (58-71) /13)      
=P(Z ≤   -1.00   ) =   0.1587   (answer)
============

X=Zσ+µ=3*13+71=   110
=P(Z ≤   3.00   ) =   0.9987   (answer)
==================

P(X≤x) =   0.0228  
      
z value at 0.0228=   -2
z=(x-µ)/σ      
so, X=zσ+µ=   -2 *13+71
X =   45 (answer)

test score	Z-score	Percentile
84	1	84.13
58	-1	15.87
110	3	99.87
45	-2	2.28