Question

In Canada, do young males (Y) weigh more than older males (O) on average? To investigate this question, weights of eighteen males randomly selected from each group were recorded. Summary statistics are given in the following table. Assume weight of each group follows a normal distribution. (If you think the samples are independent, then assume equal variances.)

Summary statistics	Young (Y)	Old (O)	Pooled	Difference
Average (kg)	82.10	75.40		6.70
Standard Deviation (kg)	16.80	14.30	15.60	5.20

What will be the respective degrees of freedom and the P-value range for testing if young Canadian males weigh more than older males on average?

Twelve inspectors, each using two different kinds of calipers, measured the diameter of a ball bearing. Summary statistics are given in the following table. Assume both samples come from normal distributions. (If you think the samples are independent, then assume equal variances.)

Summary statistics	Caliper 1	Caliper 2	Pooled	Difference
Average (mm)	0.267	0.265		0.002
Standard Deviation (mm)	0.0024	0.0022	0.0023	0.0021

What is the margin of error for a 95% confidence interval for the difference in mean diameter measurements for the two kinds of calipers?

Twelve inspectors, each using two different kinds of calipers, measured the diameter of a ball bearing. Summary statistics are given in the following table. Assume both samples come from normal distributions. (If you think the samples are independent, then assume equal variances.)

Summary statistics	Caliper 1	Caliper 2	Pooled	Difference
Average (mm)	0.267	0.263		0.004
Standard Deviation (mm)	0.0056	0.0079	0.0068	0.0041

Is there a difference in mean diameter measurements for the two kinds of calipers? In performing this hypothesis test, what is the P-value range based on the t-table and the conclusion at the 1% significance level?

Answer 1

Weights of eighteen males randomly selected from each group were recorded.

Thus n = 18

Given Summary statistics

Summary statistics	Young (Y)	Old (O)	Pooled	Difference
Average (kg)	82.10	75.40		6.70
Standard Deviation (kg)	16.80	14.30	15.60	5.20

since n = 18 ,

          = 82.10 ( Average of weight of Young )

         = 75.4    ( Average of weight of Old )

         S.E = 0.0021         ( difference in Standard deviation )

Then respective degrees of freedom d.f will be   n-1 = 17 , i.e d.f = 17

Now to find , P-value range for testing if young Canadian males weigh more than older males on average .

Mathematical probabilities like p-values range from 0 (no chance) to 1 (absolute certainty)

Here to obtain P-value for above test we need to obtain test statistics value .

Here Hypothesis will be

H0 : =              ( young Canadian males weigh are same to than older males on average )

H1 : >        ( young Canadian males weigh more than older males on average )

Test Statistics (T.S) :

T.S =

Given   S.E = 5.20

= 82.10           ,             = 75.4

Thus

T.S = =        = 1.288462

Thus P-Value is given by

P-value = Pr( X > T.S )

              = Pr( X > 1.288462 )

              = 1 - P(X < 1.288462 )

where X ~

Calculating requiter probability from R

> 1-pt( 1.288462,17)      # 1 - P(X < 1.288462 )
[1] 0.1074228

Thus

P-value = Pr( X > T.S ) = 0.10

P-value = 0.10

Q2

Twelve inspectors, each using two different kinds of calipers, measured the diameter of a ball bearing.

Summary statistics are given in the following table.

Summary statistics	Caliper 1	Caliper 2	Pooled	Difference
Average (mm)	0.267	0.265		0.002
Standard Deviation (mm)	0.0024	0.0022	0.0023	0.0021

Given S.E = 0.0021       ( Given Standard deviation of difference , hence we need not to calculate it )

To calculate margin of error for a 95% confidence interval

Thus t-value for 95% confidence interval is with n-1 = 11 degree of freedom

t-value = T.V. = 2.200985

{

> qt(1-0.05/2,11)
[1] 2.200985

}

Thus Margin of Error (M.E) is

M.E = T.V. *SE

      = 2.200985 * 0.0021      

      = 0.004622068

margin of error for a 95% confidence interval is 0.004622068.

Q3

Twelve inspectors, each using two different kinds of calipers, measured the diameter of a ball bearing.

Summary statistics are given in the following table.

Summary statistics	Caliper 1	Caliper 2	Pooled	Difference
Average (mm)	0.267	0.263		0.004
Standard Deviation (mm)	0.0056	0.0079	0.0068	0.0041

a) Is there a difference in mean diameter measurements for the two kinds of calipers?

- Hypothesis to test

H0 : =          ( there is no difference in mean diameter measurements for the two kinds of calipers)

H1 :        ( there is significant difference in mean diameter measurements )

Test Statistics :

T.S =

here

=    0.267     ,     = 0.263

S.E = 0.0041       ( Given Standard deviation of difference , hence we need not to calculate it )

Thus ,    T.S =

                     = = 0.9756098

Test Statistics : T.S = 0.9756098.

to calculate P-value

Here alternative hypothesis is of "" , hence P-value will be

P-value = Pr( X < - T.S ) + Pr( X > T.S )

              = Pr( X < - 0.9756098) + Pr( X > 0.9756098 )

              =   2 * Pr( X < - 0.9756098)

X~

from R

> pt( -0.9756098,11)          #Pr( X < - 0.9756098)
[1] 0.1751126

P-value   = 2 * Pr( X < - 0.9756098)

               = 2 * 0.1751126

P-value = 0.3502252

Thus, P-value = 0.3502252

At the 1% significance level , since P-value = 0.3502 > 0.01 , we do not reject null hypothesis at 1% of level of significance .

Hence at at the 1% significance level , we do not reject null hypothesis , and conclude that

there a may not be any difference in mean diameter measurements for the two kinds of calipers .