Question

A statistical analyst for the Wall Street Journal randomly selected six companies and recorded both the price per share of stock on January 1, 2009 and on April 30, 2009. The results are presented below. Suppose the analyst wished to see if the average price per share of stock on April 30, 2009 is greater than the average price per share of stock on January 1, 2009 at α=.05.

Apr. 30, 2009	35	38	26	29	30	34
Jan. 1, 2009	28	30	27	24	30	20

For the hypothesis stated above, what is the decision (in terms of "April 30, 2009" minus "January 1, 2009")?

	a.	Reject H0 because P-value > α
	b.	None of the answers is correct
	c.	Reject H0 because the test statistic is to the right of the positive critical value
	d.	Fail to reject H0 because the test statistic is to the right of the positive critical value
	e.	Fail to reject H0 because P-value > α

Answer 1

Given -

A statistical analyst for the Wall Street Journal randomly selected six companies and recorded both the price per share of stock on January 1, 2009 and on April 30, 2009.

The results are

Apr. 30, 2009	35	38	26	29	30	34
Jan. 1, 2009	28	30	27	24	30	20

The analyst wished to see if the average price per share of stock on April 30, 2009 is greater than the average price per share of stock on January 1, 2009 at α=.05.

Now since we wish to check average price per share of stock on April 30, 2009 is greater than the average price per share of stock on January 1, 2009 or not .

Since statistical analyst for the Wall Street Journal randomly selected six companies and recorded both the price per share of stock on January 1, 2009 and on April 30, 2009 , so here we need to use dependent sample t test

Then define   d = Price per share of stock(April 30, 2009) - Price per share of stock (January 1, 2009 )

let X_d be as estimate of d ( for difference in stocks of six companies )

To test

H0 : X_d = 0       { average price per share of stock on April 30, 2009 and on January 1, 2009 do not differ significantly }

H1 : X_d > 0    { average price per share of stock on April 30, 2009 is greater than that on January 1, 2009 }

Test statistics TS :

TS =

here n = 6

To calculate and Sd we create a table as follow

	April 30, 2009	January 1, 2009	d
1	35	28	7
2	38	30	8
3	26	27	-1
4	29	24	5
5	30	30	0
6	34	20	14

Here d = April 30, 2009 - January 1, 2009

Now = = ( 7 + 8 -1 + 5+ 0 +14 ) / 6   = 5.5

And Sd = =

                                                  =    = 5.540758

Thus = 5.5 , Sd = 5.540758

Hence

TS = = = 2.431471

Thus calculated test statistics is TS = 2.4315

To find P-value

Now alternative hypothesis is of type " > " , so this is right tail test ,

thus P-value will be given by

P-Value =Pr ( t > TS ) = P ( t > 2.4315 )

where t is is t-distributed with n-1 = 6 degree of freedom

Now P ( t > 2.4314 ) = 1 - P ( t < = 2.4315 )

It can be computed from statistical book or more accurately from any software like R,Excel

From R

> 1-pt(2.4315,df=5)          # 1 - P ( t < = 2.4315 )
[1] 0.0296355

Hence   P ( t > 2.4314 ) = 1 - P ( t < = 2.4315 ) = 0.0296355

Thus P-Value = 0.02964

Conclusion - Since P-Value = 0.02964 < 0.05 , we reject null hypothesis at α=0.05 significance level .

Hence we conclude that , the average price per share of stock on April 30, 2009 may be greater than the average price per share of stock on January 1, 2009

So Correct option is

a . Reject H₀ because P-value < α = 0.05.