Question

According to a social media​ blog, time spent on a certain social networking website has a mean of 22 minutes per visit. Assume that time spent on the social networking site per visit is normally distributed and that the standard deviation is 7 minutes. Complete parts​ (a) through​ (d) below.

a. If you select a random sample of 25 ​sessions, what is the probability that the sample mean is between 21.5 and 22.5 ​minutes? ___ ​(Round to three decimal places as​ needed.)

b. If you select a random sample of 25 sessions, what is the probability that the sample mean is between 21 and 22 minutes? ___ ​(Round to three decimal places as​ needed.)

c. If you select a random sample of 144 ​sessions, what is the probability that the sample mean is between 15.5 and 16.5 ​minutes? ​____ (Round to three decimal places as​ needed.)

d. Explain the difference in the results of​ (a) and​ (c).

The sample size in​ (c) is greater than the sample size in​ (a), so the standard error of the mean​ (or the standard deviation of the sampling​ distribution) in​ (c) is

______ than in​ (a). As the standard error _______ values become more concentrated around the mean.​ Therefore, the probability that the sample mean will fall in a region that includes the population mean will always ______ when the sample size increases.

Answer 1

a)

for normal distribution z score =(X-μ)/σx
here mean=       μ=	22
std deviation   =σ=	7.000
sample size       =n=	25
std error=σx̅=σ/√n=	1.4000

probability =

P(21.5<X<22.5)

=

P(-0.36<Z<0.36)=

0.6406-0.3594=

0.281

b)

probability =

P(21<X<22)

=

P(-0.71<Z<0)=

0.5-0.2389=

0.261

c)

sample size       =n=	144
std error=σx̅=σ/√n=	0.5833

probability =

P(15.5<X<16.5)

=

P(-11.14<Z<-9.43)=

0-0=

0.0000

d)

The sample size in​ (c) is greater than the sample size in​ (a), so the standard error of the mean​ (or the standard deviation of the sampling​ distribution) in​ (c) is smaller than in​ (a). As the standard error decrease  values become more concentrated around the mean.​ Therefore, the probability that the sample mean will fall in a region that includes the population mean will always greater _ when the sample size increases.