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- FOR A STANDARD NORMAL DISTRIBUTION,
WE WANT TO FIND C SUCH THAT THE PROBABILITY OF Z
BEING LESS THAN C EQUALS 0.614.
NOTICE HOW HERE WE HAVE THE PROBABILITY OF Z
BEING LESS THAN C
AND THEREFORE, IF WE FIND THE Z-SCORE
ON THE STANDARD NORMAL DISTRIBUTION,
THE AREA TO THE LEFT OF THAT Z-SCORE WOULD BE THE SAME
AS THIS PROBABILITY, 0.614.
AND SINCE THIS VALUE IS GREATER THAN 0.5,
WE SHOULD RECOGNIZE THAT THE Z-SCORE WILL BE POSITIVE
SINCE WHEN Z = 0, THE DATA VALUE IS THE MEAN.
THE PROBABILITY TO THE LEFT WOULD BE 50%
AND THE PROBABILITY TO THE RIGHT WOULD BE 50%.
SO IF THE PROBABILITY TO THE LEFT IS 0.614,
LET'S SAY THE Z-SCORE IS MAYBE SOMEWHERE IN HERE
WHERE THE Z-SCORE = C.
THEN THE PROBABILITY OF Z BEING LESS THAN C
WOULD BE THE SAME AS THE AREA TO THE LEFT,
WHICH WE KNOW IS 0.614.
SO THIS AREA TO THE LEFT,
WHICH IS THE SAME AS THE PROBABILITY TO THE LEFT,
IS 0.614.
SO NOW TO FIND THE VALUE OF C, OR THE Z-SCORE
THAT SATISFIES THIS PROBABILITY,
WE CAN NOW USE THE INVERSE NORM FEATURE
ON THE TI84 GRAPHING CALCULATOR.
NORMALLY WHEN USING THE INVERSE NORM FEATURE,
WE ENTER, "THE PROBABILITY TO THE LEFT, COMMA,
THE MEAN, COMMA, THE STANDARD DEVIATION."
BUT IN OUR CASE
SINCE WE'RE WORKING WITH THE STANDARD NORMAL DISTRIBUTION
AND WE'RE LOOKING FOR Z-SCORE, WE CAN LEAVE OFF MU AND SIGMA.
WHEN WE LEAVE THESE OFF,
MU = 0 AND SIGMA = 1,
WHICH ARE THE CORRECT VALUES
FOR THE STANDARD NORMAL DISTRIBUTION,
WHICH MEANS TO FIND THIS Z-SCORE OR THE VALUE OF C,
WE'LL FIRST PRESS SECOND VARS FOR THE DISTRIBUTION MENU.
THEN WE'LL SELECT OPTION THREE FOR INVERSE NORM.
AND NOW WE'LL JUST ENTER THE AREA OR PROBABILITY
TO THE LEFT,
WHICH WE KNOW IS 0.614 CLOSED PARENTHESIS AND ENTER,
AND THIS GIVES US OUR VALUE OF C OR THE Z-SCORE,
SUCH THAT THE PROBABILITY OF Z BEING LESS THAN THIS VALUE
IS EQUAL TO 0.614.
SO ROUND TO FOUR DECIMAL PLACES,
C WOULD BE APPROXIMATELY 0.2898.
I HOPE YOU FOUND THIS HELPFUL.