Question

What is the code and result in Rstudio or R

1. Suppose we have a random sample 1.12, 0.44, -1.49, 0.02, 0.81, -1.34,1.34, 0.51,-0.12, 0.97. (a). Use two methods to find the sample mean (b). Use two methods to find the sample variance. (c). Find the sample standard deviation. 2. We can use function rt(n 100, df 2) to generate a random sample from a t-distribution with two degrees of freedom. The sample size is n 100. (a). Generate a sample with 1000 observations from a t-distribution with three degrees of freedom in R, and report its 95% sample quantile. (b). Use a formal test to see whether this sample is from a normal distribu tion (c). Use QQ plot to check the normality of this sample. (d). Generate a sample with 1000 observations from a t-distribution with 100 degrees of freedom, and repeat questions (a) to (d.

Answer 1

1.(a)

R code:

x=c(1.12,0.44,-1.49,0.02,0.81,-1.34,1.34,0.51,-0.12,0.97)
S=sum(x)
M=S/length(x)# Method 1 Q1(a)
M
mean(x)# Method 2 Q1(a)
Output:

> M=S/length(x)# Method 1 Q1(a)
> M
[1] 0.226
> mean(x)# Method 2 Q1(a)
[1] 0.226

1(b)

R code:

x=c(1.12,0.44,-1.49,0.02,0.81,-1.34,1.34,0.51,-0.12,0.97)
S=sum(x)
M=S/length(x)
SS=sum(x^2)
SSW=SS-length(x)*M^2
Var=SSW/(length(x)-1)# Method 1 Q1(b)
Var
var(x)# Method 2 Q1(b)

Output:

> Var
[1] 0.9578267
> var(x)# Method 2 Q1(b)
[1] 0.9578267
1(c)

R code:

sd(x)

Output:
[1] 0.9786862

2.

(a)

R code:

x=rt(1000,2)
quantile(x,0.95)

Output:

> quantile(x,0.95)
95%
2.88103

(b)

R code:

x=rt(1000,2)
quantile(x,0.95)
shapiro.test(x)

Output:

> shapiro.test(x)

Shapiro-Wilk normality test

data: x
W = 0.87789, p-value < 2.2e-16
(c)

R code:

x=rt(1000,2)
quantile(x,0.95)
qqnorm(x)
qqline(x, col = 2)

(d)

R code:

x=rt(1000,100)
quantile(x,0.95)
shapiro.test(x)
qqnorm(x)
qqline(x, col = 2)

Output:

> quantile(x,0.95)
95%
1.561231
> shapiro.test(x)

Shapiro-Wilk normality test

data: x
W = 0.99618, p-value = 0.0147