A random sample of n=49 observations is drawn from a population with a mean equal to 49 and a standard deviation equal to 28.
a. Find the probability that x over bar x is less than 41.
b. Find the probability that x over bar x is greater than 59.
c. Find the probability that x over bar x falls between 41 and 57
a. The probability that x over bar x is less than 41 is _______(Round to three decimal places as needed.)
b. The probability that x over bar x is greater than 59 is _________. (Round to three decimal places as needed.)
c. The probability that x over bar x falls between 41 and 57 is _______. (Round to three decimal places as needed.)
Solution:- Given that mean = 49, standard deviation = 28, n = 49
a. P(X < 41) = P((X-μ)/(σ/sqrt(n)) < (41-49)/(28/sqrt(49)))
= P(Z <-2)
= 0.0228
= 0.023 (rounded)
=> The probability that x over bar x is less than 41 is 0.023
b. P(X > 59) = P(Z > (59-49)/(28/sqrt(49)))
= P(Z > 2.5)
= 0.0062
= 0.006 (rounded)
The probability that x over bar x is greater than 59 is 0.006
c. P(41 < X < 57) = P((41-49)/(28/sqrt(49)) < Z < (57-49)/(28/sqrt(49)))
= P(-2 < Z < 2)
= 0.9544
= 0.954 (rounded)
The probability that x over bar x falls between 41 and 57 is 0.954 (rounded)
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