Reaction time studies are studies in which participants receive a stimulus and t
ID: 3305057 • Letter: R
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 skewed slightly to the right. 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. a. Suppose we have 12 different people take this reaction time test. What is the probability that the average of these 12 reaction times will be greater than 184 milliseconds? b. Suppose we have 13 different people take this reaction time test. What is the probability that the average of these 13 reaction times will be less than 191 milliseconds? c. Suppose we have 23 different people take this reaction time test. What is the probability that the average of these 23 reaction times will be less than 189 milliseconds? d. Would it be appropriate to use the normal probability app to compute the probability that a single reaction time is less than 189 milliseconds? (You have two attempts for this question.) Yes, because the Central Limit Theorem makes everything become normally distributed. No, because reaction times are not normally distributed, and the normal probability app is only for computing probabilities associated with a normally distributed variable. No, because you cannot compute a probability for a single event, only for the long run relative frequency of an infinite number of events. Yes, because everything in statistics is an approximation and so it doesn't matter if our methodology makes sense. All that matters is that we use a method that produces a number of some sort. Yes, because converting a variable to a z-score makes that variable become normally distributed. No, because we can only compute the probability that reaction time is less than *or equal to* 189 milliseconds. Yes, because the Law of Large Numbers states that as a variable increases, it becomes more accurate.
Explanation / Answer
a)
Mean of the distribution is 190 milliseconds and standard deviation is 20 milliseconds.
When the Probability of reaction times of 12 people being greater than 184 milliseconds is calculated we are calculating a variable which is an average of 12 people so this variable is likely to be normally distributed.
convert x (reaction time in milliseconds) to z-variable by the formula z = (x-mu)/sigma.
P(x>=184) = P(z >= (184-190)/20)
= 1- P(z<=-0.3)
=0.6179
b)
Probability of reaction times less than 191 milliseconds
P(x<=191) = P(z <= (191-190)/20)
= P(z<=0.05)
=0.5199
c)
Probability of reaction times less than 189 milliseconds
P(x<=189) = P(z <= (189-190)/20)
= P(z<=-0.05)
=0.4801
d)
No, because reaction times are not normally distributed, and the normal probability app is only for computing probabilities associated with a normally distributed variable.
No, because you cannot compute a probability for a single event, only for the long run relative frequency of an infinite number of events.
Both of the above statements can be substantiated by saying that the reaction time as an individual event is not normally distributed, whereas when we are calculating the average of a group of reaction times then it is.
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