Question

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

7.3    7.1    6.4    7.5    7.6    6.2    6.8    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 Ha: 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 not 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 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

Here, we have given that,

Xi
7.3
7.1
6.4
7.5
7.6
6.2
6.8
7.6
6.4
7

n= Number of observation =10

(A)

= sample mean =

                             =

                             =6.99

S= sample standard deviation=

                                                =

                                                = 0.5238

Here, population standard deviation is not known, here we are using the t-test to find the confidence interval.

(B)

Now, we want to find the 99% confidence interval for population mean

The formula is as follows,

Where

E=Margin of error =

Now, first we can find the critical value

Degrees of freedom = n-1 = 10-1=9

c=confidence level =0.99

=level of significance=1-c=1-0.99=0.01

and we need to find the upper one-sided confidence bound.

t-critical = 2.821 ( using t table see value corresponding to the D.F =9 and one tailed =0.01)

Now,

E =

=

= 0.4573

We get the 99% confidence interval for the population mean

The 99% upper one-sided confidence bound for the populaiton mean is 7.457.

(C)

Claim: To check whether the populaiton mean is less than 7.5.

The null and alternative hypothesis is as follows

v/s

where, = Population mean

This is the left one-tailed test.

Now, we can find the test statistic

t-statistics =

                  =

                  =

                   = -3.079

The Test statistic is -3.079

Now we can find the critical value

= level of significance=0.01

Degrees of freedom = n-1 = 10-1=9

This is the left one-tailed test.

t-critical = 2.821 ( using t table see value corresponding to the D.F =9 and one tailed =0.01)

Decision:

Here, |t-statistics | (3.079) greater than (>) t-critical

i.e. we reject the Ho Null hypothesis.

Conclusion

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

(D)

Here, we can see that the 99% upper CI bound (7.457) is less than 7.5 i.e. based on the 99% CI also we can conclude that the mean is less than 7.5

i.e. Yes, Part (b) support the conclusion in part (C)