For each of the following studies the samples were given an

For each of the following studies, the samples were given an experimental treatment and the researchers compared their results to the general population. (Assume all populations are normally distributed.) For each, carry out a Z test using the five steps of hypothesis testing for a two-tailed test, and make a drawing of the distributions involved. ADVANCED TOPIC: Figure the 95% confidence interval for each study.

Solution

17.

FOR SAMPLE IN PART A, TWO TAILED Z TEST:

Formulating the null and alternative hypotheses,              
              
Ho:   u   =   36  
Ha:    u   =/   36  
              
As we can see, this is a    two   tailed test.      
              
Thus, getting the critical z, as alpha =    0.05   ,      
alpha/2 =    0.025          
zcrit =    +/-   1.959963985      
              
Getting the test statistic, as              
              
X = sample mean =    38          
uo = hypothesized mean =    36          
n = sample size =    16          
s = standard deviation =    8          
              
Thus, z = (X - uo) * sqrt(n) / s =    1          
              
Also, the p value is              
              
p =    0.317310508          
              
As |z| < 1.95996, and P > 0.05, we   FAIL TO REJECT THE NULL HYPOTHESIS.          

Hence, there is no significant evidence that the true mean is 36, at 0.05 level. [CONCLUSION]

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FOR SAMPLE IN PART A, CONFIDENCE INTERVAL:

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    38          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    8          
n = sample size =    16          
              
Thus,              
Margin of Error E =    3.919927969          
Lower bound =    34.08007203          
Upper bound =    41.91992797          
              
Thus, the confidence interval is              
              
(   34.08007203   ,   41.91992797   ) [ANSWER, CONFIDENCE INTERVAL]

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