Question

A random sample of n=64 observations is drawn from a population with a mean equal to 51 and a standard deviation equal to 32.

a. Find the probability that overbar x is less than 43.

b. Find the probability that overbar x is greater than 57.

c. Find the probability that x over bar x falls between 43 and 59.

​(Round to three decimal places as​ needed.)

Answer 1

sample standard deviation = (population standard deviation)/(sqrt{sample size})

sample size is n = 64 and population standard deviation = 32

this implies, sample standard deviation s = 32/sqrt{64}

= 32/8

= 4

So, sample mean = 51 and standard deviation is s = 4

(A) probability that overbar x is less than 43.

using normalcdf(lower bound, upper bound, mean, standard deviation)

setting lower bound = negative infinity or -E99, upper bound = 43, mean = 51 and standard deviation = 4

= normalcdf(-E99,43,51,4)

= 0.023

(B) probability that overbar x is greater than 57.

using normalcdf(lower bound, upper bound, mean, standard deviation)

setting lower bound =57, upper bound = infinity or E99, mean = 51 and standard deviation = 4

= normalcdf(57,E99,51,4)

= 0.067

(C) probability that x over bar x falls between 43 and 59

using normalcdf(lower bound, upper bound, mean, standard deviation)

setting lower bound =43, upper bound = 59, mean = 51 and standard deviation = 4

= normalcdf(43,59,51,4)

= 0.955