Question

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

Answer 1

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: