In: Statistics and Probability
Use R to load in the file “data.csv”. Assume that this is a random sample from some population with mean µ and variance σ 2 .
(a) Plot a histogram of the data.
(b) Compute a 95% confidence interval for the population mean µ using the formula
X ± (S/√ n)tn−1,.975.
(Hint: tn−1,.975 can be computed with qt(.975,df=n-1))
(c) Compute a p-value for the hypothesis H0 : µ = 5 versus HA : µ > 5, based on the test statistic T = X−5 S/√ n .
(Hint: The p-value can be computed using 1-pt(T,df=n-1))
(d) Based on the p-value, do we reject the null hypothesis at α = .01?
(e) What is the smallest significance level α for which we would reject the null?
Data set:
x | |
1 | 4.698166 |
2 | 4.447565 |
3 | 6.841008 |
4 | 7.013583 |
5 | 3.129358 |
6 | 5.147627 |
7 | 2.549057 |
8 | 4.061032 |
9 | 2.482377 |
10 | 6.200452 |
11 | 3.017356 |
12 | 3.54399 |
13 | 5.02652 |
14 | 5.941181 |
15 | 7.012088 |
16 | 1.780168 |
17 | 4.338341 |
18 | 8.932189 |
19 | 8.437784 |
20 | 8.858227 |
21 | 4.750132 |
22 | 9.313738 |
23 | 4.09576 |
24 | 2.746881 |
25 | 3.80401 |
26 | 9.34906 |
27 | 5.87805 |
28 | 7.306379 |
29 | 7.147015 |
30 | 4.489627 |
31 | 5.048496 |
32 | 3.97515 |
33 | 5.325467 |
34 | 8.177696 |
35 | 6.422605 |
36 | 7.81162 |
37 | 9.849941 |
38 | 9.936086 |
39 | 8.045554 |
40 | 4.141212 |
41 | 5.19843 |
42 | 6.439768 |
43 | 5.067979 |
44 | 3.790223 |
45 | 8.642296 |
46 | 10.72038 |
47 | 5.450084 |
48 | 4.960262 |
49 | 3.355154 |
50 | 4.35933 |
Sol:
> library(readr)
> Data <- read_delim("C:/Users/M1045151/Downloads/Data.csv",
+ ";", escape_double = FALSE, trim_ws = TRUE)
After importing
T get histogram in R
hist(Data$x)
From histogram we observe that x follows normal distribution
SolutionB:
Rcode:
n <- 50
qt(.975,df=50-1)
mean(Data$x)
sd(Data$x)
tc=2.009575
xbar=sample mean=5.781129
sample sd=s=
95% confidence interval for mean
xbar-tc8s/sqrt(50),xbar+tc*s/sqrt(n)
5.781129-2.009575*2.246647/sqrt(50),5.781129+2.009575*2.246647/sqrt(50)
5.142639,6.419619
95% lower limit:5.142639
95% upper limit=6.419619
SIMPLE RCODE TO GET ABOVE ANSWER:
t.test(Data$x)
Output:
One Sample t-test
data: Data$x
t = 18.195, df = 49, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
5.142639 6.419619
sample estimates:
mean of x
5.781129
95 percent confidence interval:
5.142639 6.419619
Solutionc:
Rcode:
t.test(Data$x,mu=5,alternative = "greater")
output:
One Sample t-test
data: Data$x
t = 2.4585, df = 49, p-value = 0.008766
alternative hypothesis: true mean is greater than 5
98 percent confidence interval:
5.110772 Inf
sample estimates:
mean of x
5.781129
p=0.008766
(d) Based on the p-value, do we reject the null hypothesis at α = .01?
p=0.008766
alpha=0.01
p>0.01
Fail to reject Ho.
(e) What is the smallest significance level α for which we would reject the null?
smallest level of alpha=10%=0.10
because as p=0.008766
p<alpha
reject Ho.
ANSWER:
10%
RSCREENSHOT: