Consider the following hypotheses H0 mu 88 8 HA mu 88 8 A s

Consider the following hypotheses: H_0: mu 88. 8 H_A. mu > 88. 8 A sample of 40 observations yields a sample mean of 90. 3. Assume that the sample is drawn from a normal population with a known population standard deviation of 5. 0. Use Table 1. Calculate the p-value. (Round \"z\" value to 2 decimal places. ) What is the conclusion if a = 0. 02? Calculate the p-value if the above sample mean was based on a sample of 135 observations. (Round \"z\" value to 2 decimal places. ) What is the conclusion if a = 0. 02? The screening process for detecting a rare disease is not perfect. Researchers have developed a blood test that is considered fairly reliable. It gives a positive reaction in 95. 5% of the people who have that disease. However, it erroneously gives a positive reaction in 5. 0% of the people who do not have the disease. Answer the following questions using the null hypothesis as \"the individual does not have the disease. \" What is the probability of Type I error? (Round your answer to 3 decimal places. ) What is the probability of Type II error? (Round your answer to 3 decimal places. )

Solution

A)

Formulating the null and alternative hypotheses,              
              
Ho:   u   <=   88.8  
Ha:    u   >   88.8  
              
As we can see, this is a    right   tailed test.      
              
              
Getting the test statistic, as              
              
X = sample mean =    90.3          
uo = hypothesized mean =    88.8          
n = sample size =    40          
s = standard deviation =    5          
              
Thus, z = (X - uo) * sqrt(n) / s =    1.897366596 = 1.90          
              
Thus, the p value is              
              
p = 0.02871656 [ANSWER]  

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B)

As P > 0.02, we   DO NOT REJECT THE NULL HYPOTHESIS.   [ANSWER]      

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C)

Getting the test statistic, as              
              
X = sample mean =    90.3          
uo = hypothesized mean =    88.8          
n = sample size =    135          
s = standard deviation =    5          
              
Thus, z = (X - uo) * sqrt(n) / s =    3.485685012 = 3.49          
              
Also, the p value is              
              
p = 0.00024151 [answer]

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D)
              
As P < 0.02, we   REJECT THE NULL HYPOTHESIS.   [ANSWER]     

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