Question

Are medical students more motivated than law students? A randomly selected group of each were administered a survey of attitudes toward Life, which measures motivation for upward mobility. The scores are summarized below. The researchers suggest that there are occupational differences in mean testosterone level. Medical doctors and university professors are two of the occupational groups for which means and standard deviations are recorded and listed in the following table.

Group	Sample size	Mean	StDev
Medical	n1=4n1=4	x¯1=87.01x¯1=87.01	s1=6.7s1=6.7
Law	n2=10n2=10	x¯2=85.95x¯2=85.95	s2=16.5s2=16.5

Let us denote:

• μ1:μ1: population mean testosterone among medical doctors,
• μ2:μ2: population mean testosterone among university professors,
• σ1:σ1: population standard deviation of testosterone among medical doctors,
• σ2:σ2: population standard deviation of testosterone among university professors.

If the researcher is interested to know whether the mean testosterone level among medical doctors is higher than that among university professors, what are the appropriate hypotheses he should test?
H0:μ1=μ2H0:μ1=μ2   against   Ha:μ1≠μ2Ha:μ1≠μ2.
H0:x¯1=x¯2H0:x¯1=x¯2   against   Ha:x¯1>x¯2Ha:x¯1>x¯2.
H0:x¯1=x¯2H0:x¯1=x¯2   against   Ha:x¯1<x¯2Ha:x¯1<x¯2.
H0:μ1=μ2H0:μ1=μ2   against   Ha:μ1>μ2Ha:μ1>μ2.
H0:x¯1=x¯2H0:x¯1=x¯2   against   Ha:x¯1≠x¯2Ha:x¯1≠x¯2.
H0:μ1=μ2H0:μ1=μ2   against   Ha:μ1<μ2Ha:μ1<μ2.

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Case 1: Assume that the population standard deviations are unequal, i.e. σ1≠σ2σ1≠σ2.
What is the standard error of the difference in sample mean x¯1−x¯2x¯1−x¯2? i.e. s.e.(x¯1−x¯2)=s.e.(x¯1−x¯2)= [answer to 4 decimal places]

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Rejection region: We reject H0H0 at 1% level of significance if:
t<−3.05t<−3.05.
t>3.05t>3.05.
t<−2.68t<−2.68.
t>2.68t>2.68.
|t|>3.05|t|>3.05.
None of the above.

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The value of the test-statistic is: Answer to 3 decimal places.

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If α=0.01α=0.01, and the p-value is 0.4335, what will be your conclusion?
There is not enough information to conclude.
Do not reject H0H0.
Reject H0H0.

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Case 2: Now assume that the population standard deviations are equal, i.e. σ1=σ2σ1=σ2.
Compute the pooled standard deviation, spooledspooled [answer to 4 decimal places]

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Rejection region: We reject H0H0 at 1% level of significance if:
t>3.05t>3.05.
t<−2.68t<−2.68.
t>2.68t>2.68.
|t|>3.05|t|>3.05.
t<−3.05t<−3.05.
None of the above.

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The value of the test-statistic is: Answer to 3 decimal places.

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If α=0.01α=0.01, , and the p-value is 0.4525, what will be your conclusion?
Reject H0H0.
Do not reject H0H0.
There is not enough information to conclude.

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Answer 1

H0:μ1=μ2 against   Ha:μ1>μ2

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Sample #1   ---->     
mean of sample 1,    x̅1=   87.01          
standard deviation of sample 1,   s1 =    6.7          
size of sample 1,    n1=   4          
                  
Sample #2   ---->        
mean of sample 2,    x̅2=   85.950          
standard deviation of sample 2,   s2 =    16.50          
size of sample 2,    n2=   10          
                  
difference in sample means = x̅1-x̅2 =    87.010   -   85.9500   =   1.0600
                  
std error , SE =    √(s1²/n1+s2²/n2) =    6.2006          
----------------------------------

α=0.01

t>2.68

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t-statistic = ((x̅1-x̅2)-µd)/SE = (   1.0600   /   6.2006   ) =   0.1710

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p-value>α , Do not reject null hypothesis

There is not enough information to conclude that the mean testosterone level among medical doctors is higher than that among university professors

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case II

ASSUMING equal variance

Degree of freedom, DF=   n1+n2-2 =    12
t-critical value , t* =        2.6810

t>2.68

difference in sample means =    x̅1-x̅2 =    87.0100   -   86.0   =   1.06  
                          
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    14.68                  
std error , SE =    Sp*√(1/n1+1/n2) =    8.6829                 
                          
t-statistic = ((x̅1-x̅2)-µd)/SE = (   1.0600   -   0   ) /    8.68   =   0.122

   0.4524
p-value>α , Do not reject null hypothesis  

There is not enough information to conclude that the mean testosterone level among medical doctors is higher than that among university professors