Question

The following n = 10 observations are a sample from a normal population.

7.3    7.0    6.4    7.4    7.6    6.3    6.9    7.6    6.4    7.0

(a) Find the mean and standard deviation of these data. (Round your standard deviation to four decimal places.)

mean
standard deviation

(b) Find a 99% upper one-sided confidence bound for the population mean μ. (Round your answer to three decimal places.)


(c) Test H₀: μ = 7.5 versus H_a: μ < 7.5. Use α = 0.01.

State the test statistic. (Round your answer to three decimal places.)

t =

State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.)

t >

t <

State the conclusion.

H₀ is rejected. There is insufficient evidence to conclude that the mean is less than 7.5.

H₀ is not rejected. There is sufficient evidence to conclude that the mean is less than 7.5.    

H₀ is not rejected. There is insufficient evidence to conclude that the mean is less than 7.5.

H₀ is rejected. There is sufficient evidence to conclude that the mean is less than 7.5.

(d) Do the results of part (b) support your conclusion in part (c)?

Yes

No   

Answer 1

SolutioNA:

create a vector smpl and use mean and sd functions

smpl <- c(7.3 , 7.0 , 6.4 , 7.4 , 7.6 , 6.3,
6.9 , 7.6 , 6.4 , 7.0)
mean(smpl)
sd(smpl)

Output:

mean(smpl)
[1] 6.99
> sd(smpl)
[1] 0.4931757

MEAN=6.99

STANDARD DEVIATION=0.49318

Solutionb:

t.test(smpl,conf.level = 0.99)

output:

One Sample t-test

data: smpl
t = 44.82, df = 9, p-value = 6.849e-12
alternative hypothesis: true mean is not equal to 0
99 percent confidence interval:
6.483169 7.496831
sample estimates:
mean of x
6.99

UPPER LIMIT=7.497

Solutonc:

R code is:

t.test(smpl,mu=7.5,alternative = "less",conf.level = 0.99)

One Sample t-test

One Sample t-test

data: smpl

t = -3.2702, df = 9, p-value = 0.00484

alternative hypothesis: true mean is less than 7.5

99 percent confidence interval:

-Inf 7.43002

sample estimates:

mean of x

6.99

test statistic=t=-3.270

critical t<-2.821

p=0.00484

p<0.01

Reject h0

sufficient evidence to support the claim.

H0 is rejected. There is sufficient evidence to conclude that the mean is less than 7.5.

Solutiond:

YES