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 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 10 different people take this reaction time test. What is the probability that the average of these 10 reaction times will be greater than 182 milliseconds? b. Suppose we have 15 different people take this reaction time test. What is the probability that the average of these 15 reaction times will be less than 194 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 188 milliseconds? d. Would it be appropriate to use the normal probability app to compute the probability that a single reaction time is less than 188 milliseconds? (You have two attempts for this question.) Yes, because converting a variable to a z-score makes that variable 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 Yes, because the Law of Large Numbers states that as a variable increases, it becomes more accurate Yes, becauethe Central Limit Theorem makes everything become normally distributed. No, because you cannot compute a probability for a single event, only for the long run relative frequency of an infinite number of events. No, because we can only compute the probability that reaction time is less than -or equal to 188 milliseconds 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.

Explanation / Answer

a) P(X > 182) = P(X - mean)/(sd/sqrt(n)) > (182 - mean)/(sd/sqrt(n))

                       = P(Z > (182 - 190)/(20/sqrt(10)))

                        = P(Z > -1.26)

                        = 1 - P(Z < -1.26)

                        = 1 - 0.1038

                        = 0.8962

b) P(X < 194) = P(X - mean)/(sd/sqrt(n)) < (194 - mean)/(sd/sqrt(n))

                       = P(Z < (194 - 190)/(20/sqrt(15)))

                      = P(Z < 0.77) = 0.7794

c) P(X < 188) = P(X - mean)/(sd/sqrt(n)) < (188 - mean)/(sd/sqrt(n))

                       = P(Z < (188 - 190)/(20/sqrt(22)))

                      = P(Z < -0.47 ) = 0.3192

d) Yes, because central limit theorem makes everything normally distributed.

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