Question

***Please solve the problem below using R, and provide the R code you used in order to do so***

The mean undergraduate cost for tuition, fees, room, and board for four-year institutions was $26,489 for the 2004-2005 academic year. Suppose that the standard deviation of the distribution was $3204 and that 36 four-year institutions are randomly selected. Find the probability that the sample mean cost for these 36 schools is

a. Less than $25,000

b. Greater than $26,000

c. Between $24,000 and $26,000

Answer 1

Solution:

Given:  The mean undergraduate cost for tuition, fees, room, and board for four-year institutions was $26,489 for the 2004-2005 academic year.

thus Mean =

u=26489

the standard deviation of the distribution was $3204

thus standard deviation =

0 = 3204

Sample size = n = 36

We have to find:

Part a) P(  the sample mean cost for these 36 schools is less than $25,000) =..............?

Pla <25000) = ........

We use following R code to find this probability:

pnorm( x , mean=_____, sd=_____/sqrt(n))

pnorm(25000, mean=26489, sd=3204/sqrt(36))

0.002648541

that is:

P T < 25000) = 0.0026

Part b)

P(  the sample mean cost for these 36 schools is  Greater than $26,000) =............?

PT> 26000) = ........

pnorm(26000, mean=26489, sd=3204/sqrt(36),lower.tail=FALSE)

0.8200958

thus

PT> 26000) = 0.8200958

PI > 26000) = 0.8201

Part c)

P(  the sample mean cost for these 36 schools is Between $24,000 and $26,000) =...........?

P(24000 < i < 26000) = .........

We use R code:

pnorm(26000, mean=26489, sd=3204/sqrt(36))-pnorm(24000, mean=26489, sd=3204/sqrt(36))

0.1799026

thus

P(24000 < i < 26000) = 0.1799026

P(24000 < i < 26000) = 0.1799

R RGui (32-bit) - [R Console] R File Edit View Misc Packages Windows Help Platform: i386-w64-mingw32/i386 (32-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. Natural language support but running in an English locale R is a collaborative project with many contributors. Type 'contributors ()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or "help.start()' for an HTML browser interface to help. Type '()' to quit R. [Previously saved workspace restored] > pnorm (25000, mean=26489, sd=3204/ sqrt (36)) [1] 0.002648541 > pnorm (26000, mean=26489, sd=3204/sqrt (36), lower.tail=FALSE) [1] 0.8200958 > pnorm (26000, mean=26489, sd=3204/sqrt (36))-pnorm (24000, mean=26489, sd=3204/sqrt (36)) [1] 0.1799026