Answer:
(a) Null Hypothesis,  :
 :  = $1,150
 = $1,150  
     Alternate Hypothesis,  :
 :  
  $1,150
 $1,150 
(b) The test statistic is 3.571.
(c) We conclude that the mean of all account balances is significantly different from $1,150.
Step-by-step explanation:
We are given that a sample of 81 account balances of a credit company showed an average balance of $1,200 with a standard deviation of $126.
We have test the hypothesis to determine whether the mean of all account balances is significantly different from $1,150.
<u><em>Let </em></u> <u><em> = mean of all account balances</em></u>
<u><em> = mean of all account balances</em></u>
(a)So, Null Hypothesis,  :
 :  = $1,150     {means that the mean of all account balances is equal to $1,150}
 = $1,150     {means that the mean of all account balances is equal to $1,150}
Alternate Hypothesis,  :
 :  
  $1,150    {means that the mean of all account balances is significantly different from $1,150}
 $1,150    {means that the mean of all account balances is significantly different from $1,150}
The test statistics that will be used here is <u>One-sample t test statistics</u> as we don't know about population standard deviation;
                                T.S.  =  ~
  ~ 
where,  = sample average balance = $1,200
 = sample average balance = $1,200
              s = sample standard deviation = $126
              n = sample of account balances = 81
(b) So, <u>test statistics</u>  =   ~
  ~ 
                                =  3.571
The value of the sample test statistics is 3.571.
(c) <u>Now, P-value of the test statistics is given by the following formula;</u>
           P-value = P(  > 3.571) = <u>Less than 0.05%</u>
 > 3.571) = <u>Less than 0.05%</u>
Because in the t table the highest critical value for t at 80 degree of freedom is given between 3.460 and 3.373 at 0.05% level.
Now, since P-value is less than the level of significance as 5% > 0.05%, so we sufficient evidence to reject our null hypothesis.
Therefore, we conclude that the mean of all account balances is significantly different from $1,150.