Question

*Answer all questions using R-Script*

Question 1

Using the built in CO2 data frame, which contains data from an experiment on the cold tolerance of Echinochloa crus-galli; find the following.

a) Assign the uptake column in the dataframe to an object called "x"

b) Calculate the range of x

c) Calculate the 28th percentile of x

d) Calculate the sample median of x

e) Calculate the sample mean of x and assign it to an object called "xbar"

f) Calculate the interquartile range of x

g) Calculate the sample standard deviation of x and assign it to an object called "s"

h) Find the lower boundary for a 94 percent confidence interval for μ.

i) calculate the critical value for a 93 percent confidence interval for μ.

j) Find the upper boundary for a 94 percent confidence interval for μ.

k) how long is the 94 percent confidence interval for μ?

Question 2

Suppose A is a standard normal random variable and B is a normal random variable with mean 8 and standard deviation 11. Find the following:

a) Calculate the 29th percentile of the distribution of A

b) Calculate the 29th percentile of the distribution of B

c) What is the expected value of the random variable 7B-5

d) What is the standard deviation of the random variable 7B-5

e) What is the standard deviation of the random variable B*(-7/11)

Question 3

Let X ₁, X ₂, ... , X ₁₂ come from a sample of the normal distribution. Suppose the sample mean is x and the sample standard deviation is s. With this information, we wish to test the null hypothesis H0:μ=35 against the alternative Ha:μ<35. Out test statistic will be (x-35)/(s*sqrt(12)). Our sample yields x = 33.15 and s = 1.96. Answer the following:

a) If xbar = 33.15 and s = 1.96 then what is the value of T?

b) How many degrees of freedom does this distribution have if T has a t distribution

c) If we test at the 2% level, then what is the critical value?

d) What is the p-value based upon our sample?

e) Do we reject the null hypothesis? (yes or no)

f) If we repeat a 2 % level test 400 times, about how many times do we expect to commit a type I error?

Question 4

Say we have two independent normal populations X and Y with known standard deviations σ_x = 5.1 and σ_y = 5.3 . We then take a random sample of size 7 from X (X₁, X₂, … X₇) and a random sample from y of size 5 as follows

X: 10.75, 12.05, 2.25, 1.26, -1.43, 1.69, 15.28
Y: 6.95, 8.22, 7.95, 5.85, -2.44

Calculate the following using R:

a) Calculate xbar

b) calculate the variance of xbar

c) calculate ybar

d) calculate the variance of ybar

e) calculate the variance of xbar-ybar

f) What is the critical value used for a 95% confidence interval for μ_x - μ_y?

g) create a 95 % confidence interval for μ_x - μ_y.

h) what is the length of the 95% confidence interval for μ_x - μ_y?

i) What would the p value have been if we used this data to test H₀:μ_x - μ_y=0 against the alternative H_a:μ_x - μ_y > 0?

Question 5

We desire to know the probability that a voter supports a controversial rose proposal. From a random sample of 3100, x =1331 voters support the rose proposal. Let phat be the sample proportion supporting the rose proposal. Answer the following:

a) what is the variance of p hat

b) As a function of p, what is the standard deviation of p hat

c) Calculate p hat

d) Let ptot be the random variable representing the voters in the sample who support the rose proposal. As a function of p, what is the standard deviation of ptot?

e) Calculate a classical 96% confidence interval for p

f) What is the critical value for an approximate classical 94 percent confidence interval for p?

g) What is the length of the 96% confidence interval for p?

h) Assuming the same phat value, what sample size would have made the 96% confidence interval for p to have a length of .08 or less?

Answer 1

Question 1

All R scripts are shown in bold

a) Assign the uptake column in the dataframe to an object called "x"

x = CO2$uptake

b) Calculate the range of x

range(x)
[1] 7.7 45.5

c) Calculate the 28th percentile of x

quantile(x, 0.28)
28%
18.292

d) Calculate the sample median of x

median(x)
[1] 28.3

e) Calculate the sample mean of x and assign it to an object called "xbar"

xbar = mean(x)

f) Calculate the interquartile range of x

IQR(x)
[1] 19.225

g) Calculate the sample standard deviation of x and assign it to an object called "s"

s = sd(x)

h) Find the lower boundary for a 94 percent confidence interval for μ.

For 94 percent confidence interval, = 1 - 0.94 = 0.06

Thus, the boundary intervals are 0.06/2 and 1 - 0.06/2 = 0.03 and 0.97

Lower limit = Mean - z * standard error

where Standard error = s / sqrt(n) where n is sample size

and z is z value for 94 percent confidence interval.

So, the R script is,

xbar - (qnorm(0.97) * s / sqrt(length(x)))
[1] 24.99385

i) calculate the critical value for a 93 percent confidence interval for μ.

For 93 percent confidence interval, = 1 - 0.93 = 0.07

Thus, the boundary intervals are 0.07/2 and 1 - 0.07/2 = 0.035 and 0.965

The crirtical value of z for 93 percent confidence interval is

qnorm(0.965)

[1] 1.811911

Lower and upper limit of the confidence interval are

xbar - (qnorm(0.965) * s / sqrt(length(x)))
[1] 25.07513
xbar + (qnorm(0.965) * s / sqrt(length(x)))
[1] 29.35106

j) Find the upper boundary for a 94 percent confidence interval for μ.

xbar + (qnorm(0.97) * s / sqrt(length(x)))

[1] 29.43234

k) how long is the 94 percent confidence interval for μ?

Calculate the lower and upper limit and then calculate their difference for length of the confidence interval.

ll = xbar - (qnorm(0.97) * s / sqrt(length(x)))
ul = xbar + (qnorm(0.97) * s / sqrt(length(x)))
ul - ll
[1] 4.438482