Reaction time studies are studies in which participants receive a stimulus and t

Question

Reaction time studies are studies in which participants receive a stimulus and the amount of time it takes for them to react is measured. In one simple type of reaction time study, each participant holds a clicker button and stares at a screen. When the participant sees a part of the screen light up, he or she clicks the button as quickly as possible. The researcher then records how much time elapsed between when the screen lit up and when the participant clicked the button. Suppose that, in these tests, the distribution of reaction times is normal. Suppose also that mean reaction time is 190 milliseconds, and the standard deviation for reaction times is 20 milliseconds (for the purposes of this problem, you can treat the mean and standard deviation as population parameters). Use this information to answer the following questions, and round your answers to four decimal places. Suppose we have 9 different people take this reaction time test. What is the probability that the average of these 9 reaction times will be greater than 183 milliseconds? Suppose we have 17 different people take this reaction time test. What is the probability that the average of these 17 reaction times will be between 187 and 192 milliseconds? Suppose we have 22 different people take this reaction time test. What is the probability that the average of these 22 reaction times will be less than 185 milliseconds?

Explanation / Answer

a)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    183      
u = mean =    190      
n = sample size =    9      
s = standard deviation =    20      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -1.05      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -1.05   ) =    0.853140944 [ANSWER]

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b)

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    187      
x2 = upper bound =    192      
u = mean =    190      
n = sample size =    17      
s = standard deviation =    20      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -0.618465844      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    0.412310563      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.268134153      
P(z < z2) =    0.659944096      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.391809943   [ANSWER]

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c)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    185      
u = mean =    190      
n = sample size =    22      
s = standard deviation =    20      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -1.17260394      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -1.17260394   ) =    0.120477334 [ANSWER]
  

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