A recent study focused on the number of times men and women

A recent study focused on the number of times men and women send a Twitter message in a day. The information is summarized below. At the 0.01 significance level, is there a difference in the mean number of times men and women send a Twitter message in a day?  You may assume the population standard deviations are not the same. (Ugly df = 37 if needed.)

Solution

Formulating the null and alternative hypotheses,              
              
Ho:   u1 - u2   =   0  
Ha:   u1 - u2   =/   0  
At level of significance =    0.01          
As we can see, this is a    2   tailed test.      
Calculating the means of each group,              
              
X1 =    22          
X2 =    18          
              
Calculating the standard deviations of each group,              
              
s1 =    5          
s2 =    10          
              
Thus, the standard error of their difference is, by using sD = sqrt(s1^2/n1 + s2^2/n2):              
              
n1 = sample size of group 1 =    25          
n2 = sample size of group 2 =    30          
Thus, df = n1 + n2 - 2 =    53          
Also, sD =    2.081665999          
              
Thus, the t statistic will be              
              
t = [X1 - X2 - uD]/sD =    1.921537846          
              
where uD = hypothesized difference =    0          
              
Now, the critical value for t is              
              
tcrit =    2.715408722          
              
Thus, comparing t and tcrit, we decide to   WE FAIL TO REJECT THE NULL HYPOTHESIS.          
              
Also, using p values,              
              
p =    0.060048416          

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Thus, it is OPTION A: Fail to reject the null hypothesis and conclude the means are the same. [ANSWER, OPTION A]