Question

The length of time a bat stays in a cave has a normal distribution with ? = 95 days and a standard deviation of ?=5.7 days. Suppose that we measure 445 different bats for time in a cave. Let X be the random variable representing the mean number of days and let Xtot be the random variable representing the sum of the days in a cave of the 445 examined bats.

a) About what proportion of bats stay between 90 and 100 days in a cave?

b) About proportion of bats stay fewer than 85 days in a cave?

c) About how many of the 445 bats stayed fewer than 85 days in a cave? (nearest integer)

d) About how many of the 445 bats stayed between 90 and 100 days in a cave?(nearest integer)

e) What is the standard deviation of the distribution of X?

f) What is the standard deviation of the distribution of Xtot?

g) What is the probability that 94.5 < X < 95.5?

h) What is the probability that 42,200 < Xtot < 42,800?

f) Copy your R script for the above into the text box here.

USE R CODE IF POSSIBLE // NOT MANDATORY

Answer 1

a) About what proportion of bats stay between 90 and 100 days in a cave?

R code to get proportion of bats stay between 90 and 100 days in a cave is,

mean = 95
std.dev = 5.7
n = 445
std.err = std.dev / sqrt(n) # Standard error of the mean
# a.
z.upper = (100 - mean)/std.dev
z.lower = (90 - mean)/std.dev
ans.a = pnorm(z.upper) - pnorm(z.lower)

The output of the code is 0.6196182

b) About proportion of bats stay fewer than 85 days in a cave?

R code to get proportion of bats stay fewer than 85 days in a cave is,

# b.
z = (85 - mean)/std.dev
ans.b = pnorm(z)

The output of the code is 0.0396822

c) About how many of the 445 bats stayed fewer than 85 days in a cave? (nearest integer)

Number of bats stayed fewer than 85 days in a cave is

ans.c = round(n * pnorm(z))

The output of the code is 18.

d) About how many of the 445 bats stayed between 90 and 100 days in a cave?(nearest integer)

Number of bats stayed between 90 and 100 days in a cave is

ans.d = round(n * (pnorm(z.upper) - pnorm(z.lower)))

The output of the code is 276.

e) What is the standard deviation of the distribution of X?

std.err = std.dev / sqrt(n) # Standard error of the mean

The output of the code is 0.2702059

f) What is the standard deviation of the distribution of Xtot?

std.tot = sqrt(n*std.dev^2) #standard deviation of total

The output of the code is 120.2416

g) What is the probability that 94.5 < X < 95.5?

z.upper = (95.5 - mean)/std.err
z.lower = (94.5 - mean)/std.err
ans.g = pnorm(z.upper) - pnorm(z.lower)

The output of the code is 0.9357499

h) What is the probability that 42,200 < Xtot < 42,800?

z.upper = (42800 - n* mean)/std.tot
z.lower = (42200 - n * mean)/std.tot
ans.h = pnorm(z.upper) - pnorm(z.lower)
The output of the code is 0.7335958
f) Copy your R script for the above into the text box here.

All R codes are given below with screenshot.

mean = 95
std.dev = 5.7
n = 445
# a.
z.upper = (100 - mean)/std.dev
z.lower = (90 - mean)/std.dev
ans.a = pnorm(z.upper) - pnorm(z.lower)
# b.
z = (85 - mean)/std.dev
ans.b = pnorm(z)
# c.
ans.c = round(n * pnorm(z))
# d.
ans.d = round(n * (pnorm(z.upper) - pnorm(z.lower)))
#e.
std.err = std.dev / sqrt(n) # Standard error of the mean
#f.
std.tot = sqrt(n*std.dev^2) #standard deviation of total
# g.
z.upper = (95.5 - mean)/std.err
z.lower = (94.5 - mean)/std.err
ans.g = pnorm(z.upper) - pnorm(z.lower)
# h.
z.upper = (42800 - n* mean)/std.tot
z.lower = (42200 - n * mean)/std.tot
ans.h = pnorm(z.upper) - pnorm(z.lower)

1 mean95 2 std.dev=5.7 n = 445 #a. 5 6 z, upper = (100-mean)/std. dev z.lower = (90-mean)/std.dev 7 ans.a pnorm(z.upper) pnorm(z.lower) 8 #b 9 z = (85 -mean)/std.dev 10 ans.b pnorm(z) 11 #c. 12 ans.c-round(n * pnorm(z)) 13 #d 14 ans.d-round(n*(pnorm(z.upper) - pnorm(z.lower))) 15 16 std.err = std.dev / sqrt(n) 17粁. 18 std.tot-sqrt(n*std.dev^2) #standard deviation of total 19 #g 20 z. upper = (95.5-mean)/std. err 21 z.lower = (94.5 -mean)/std.err 22 ans.g pnorm(z.upper) pnorm(z.lower) 23 #h 24 z. upper = (42800- n-mean)/std. tot 25 z.lower = (42200-n * mean)/std.tot 26 ans.h-pnorm(z.upper) pnorm(z.lower) # Standard error of the mean