A researcher is interested in whether people are able to identify emotions corre

Question

A researcher is interested in whether people are able to identify emotions correctly when they are extremely tired. It is known that, using a particular method of measurement, the accuracy ratings of people in the general population (who are not extremely tired) are normally distributed with a mean of 82 and a variance of 20. In the present study, however, the researcher arranges to test 50 people who had no sleep the previous night. The mean accuracy for these 50 individuals was 78. Using the .05 level, what should the researcher conclude? Carry out a Z test using the five steps of hypothesis testing. Make a drawing of the distributions involved. Explain your answer to someone who knows about hypothesis testing with a sample of a single individual but who knows nothing about hypothesis testing with a sample of more than one individual. ADVANCED TOPIC: Figure the 95% confidence interval and explain your answer to someone who is familiar with the general logic of hypothesis testing, the normal curve, Z scores, probability, and the idea of a distribution of means, but who has not heard of confidence intervals.

Explanation / Answer

d)

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 =    78          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    4.472135955          
n = sample size =    50          
              
Thus,              
Margin of Error E =    1.239590065          
Lower bound =    76.76040994          
Upper bound =    79.23959006          
              
Thus, the confidence interval is              
              
(   76.76040994   ,   79.23959006   ) [ANSWER]

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This means that if we repeat the same experiment many times, we can actually "trap" the true mean inside our confidence interval 95% of the time. We got this confidence level using the normal distribution, because sample means are normally distrbuted in this question. [CONCLUSION]

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