Question

1. A machine produces metal rods used in an automobile suspension system. A random sample of 12 rods is selected and the diameter is measured. The resulting data, in mm, are shown here:

                                                                                               

3.11     2.88     3.08     3.01

2.84     2.86     3.04     3.09

3.08     2.89     3.12     2.98

a.         Find a two-sided 95% confidence interval for the mean rod diameter. State the assumption necessary to find the confidence interval.  

2. Is there any evidence to indicate that mean rod diameter is different from 2.95 at significance level a = 0.05? Write the appropriate hypothesis. Make required assumption(s).
3. Can you make the same decision as in part b. from the result of part a. without testing the Hypothesis? If yes, how?

Answer 1

n = 12

Sample mean:

(Round to 3 decimal)

Sample standard deviation:

s = 0.1050 (Round to 4 decimal)

a)

Assumptions for confidence interval:

i) Data must be from a random sample from large population.

ii) Observations in the sample must be independent of each other.

Here population standard deviation is not known so we use t interval.

Confidence level = c = 0.95

95% confidence interval for the mean rod diameter is

where tc is t critical value for c = 0.95 and degrees of freedom = n - 1 = 12 - 1 = 11

tc = 2.201 (From statistical table of t values)

(Round to 4 decimal)

95% confidence interval for the mean rod diameter is (2.9313, 3.0647)

b)

Assumptions for hypothesis testing:

i) Data must be from a random sample from large population.

ii) Observations in the sample must be independent of each other.

Here population standard deviation is not known so we use t interval.

Here we have to test that

Null hypothesis :

Alternative hypothesis :

where

Test statistic:

t = 1.584 (Round to 3 decimal)

Test statistic = t = 1.584

alpha = significance level = 0.05

Degrees of freedom = n - 1 = 12 - 1 = 11

Test is two tailed test.

P value from excel using function:

=T.DIST.2T(1.584,11)

= 0.1415

P value = 0.1415

Here p value > alpha

So we fail to reject H0.

Conclusion: There is insufficient evidence to indicate that mean rod diameter is different from 2.95

We can make the same decision as in part b) from the result of part a) without testing the Hypothesis.

Because test is two tailed test and we have calculated confidence interval for two sided.

95% confidence interval for the mean rod diameter is (2.9313, 3.0647)

Confidence interval contains null value 2.95.

So we fail to reject H0.

Conclusion: There is insufficient evidence to indicate that mean rod diameter is different from 2.95