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Given a normal distribution with mean = 51 and Standard deviation = 8, and given you select a sample of n = 100, complete parts (a) through (d).
a. What is the probability that X is less than 49?
P(X <49) =
(Type an integer or decimal rounded to four decimal places as needed.)
b. What is the probability that X is between 49 and 51.5?
P(49 < X < 51.5) =
(Type an integer or decimal rounded to four decimal places as needed.)
c. What is the probability that X is above 51.5?
P(X > 51.5) =
(Type an integer or decimal rounded to four decimal places as needed.)
d. There is a 35% chance that X is above what value?
X=
(Type an integer or decimal rounded to two decimal places as needed.)
~~~~~~~~~~~~~~~~~
All 4 (four) answers in the post by @Theo are incorrect,
since in his calculations he mistakenly used Standard Deviation SD = 0.8
(as you can see from his snapshots), while the given value to use is SD= 8.
The correct answers are (I used for calculations regular calculator TI84,
but you can use the same online calculator https://davidmlane.com/hyperstat/z_table.html
as Theo used, too - it is on your choice).
z1 z2 mean SD <<<---=== formatting pattern
(a) P = normalcdf(-9999, 49, 51, 8) = 0.4013.
z1 z2 mean SD <<<---=== formatting pattern
(b) P = normalcfd( 49, 51.5, 51, 8) = 0.1236.
z1 z2 mean SD <<<---=== formatting pattern
(c) P = normalcfd( 51.5, 9999, 51, 8) = 0.4751.
These three calculations compute the area under the normal curve (standard function normalcfd).
The last, fourth calculation computes z-score from the given area (standard function invNorm).
area mean SD <<<---=== formatting pattern
(d) z = invNorm(0.35, 51, 8) = 54.083.
Solved.
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Notice that in the condition, these words "and given you select a sample of n = 100" are
to the problem, came there by of the problem's composer,
and should be from there, in order for the problem makes sense.
