Question

A study is conducted to find out whether the wait times at two local banks are different. The sample statistics are listed below. Test whether the wait times are the same or different assuming that 0,=02. Use a = 0.05. Bank 1 Bank 2 ni = 15 n2 = 16 Ij = 5.3 minutes T2 = 5.6 minutes $1 1.1 minutes 82 1.0 minutes

Answer 1

in above question , we need to test

To Test :-

H0 : u1 = u2               [ wait time at two local bank do not differ significantly , i.e they are same ]

H1 : u1 $\neq$ u2              [ wait time at two local bank differ significantly , i.e they are not same ]

Given level of significance $\alpha$ = 0.05 , and we assume equal variance i.e $\sigma$ 1 = $\sigma$ 2

Test Statistics TS :-

TS =

...1 .x2 SE

where

S.E =

sda/n1 + sda/n2

and $sd^2$ = $\frac{(n1-1)*s1^{2}+(n2-1)*s2^{2}}{n1+n2-2}$

From Above given data we have

   $sd^2$ = $\frac{(n1-1)*s1^{2}+(n2-1)*s2^{2}}{n1+n2-2}$

           =

(15-1)* 1.1" + (16-1) + 1.02 15+ 16-2

   $sd^2$ =1.101379           { after calculation }

Thus

S.E =

sda/n1 + sda/n2

       = $\sqrt{1.101379 /15 +1.101379 /16 }$

S.E = 0.3771756

hence we have

TS =

...1 .x2 SE

     = $\frac{5.3-5.6}{0.3771756}$

TS = -0.7953855

So , calculated test statistics value is TS = -0.7953855

Rejection criteria :-

We reject null hypothesis if absolute value of calculated test statistics is greater than t-critical value $t_{\alpha /2}$

i.e Reject H0 if | TS | > $t_{\alpha /2}$

where $t_{\alpha /2}$ is t-distributed with n1+n2-2 = 15+16-2 = 29 degree of freedom at $\alpha$ = 0.05

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

From R

> qt(1-0.05/2,df=29)
[1] 2.04523

Thus t-critical value $t_{\alpha /2}$ = 2.04523

Now | TS |= | -0.7953855 | = -0.7953855   < 2.04523

i.e   | TS | < $t_{\alpha /2}$

So we fail to reject null hypothesis at 5% of level of significance .

Conclusion :-

Since we do not have enough evidence against null hypothesis H0 , so we conclude that wait time at two local bank may not be different