Question

check your understanding
on a website that has brain games and cognitive tests, the distribution of reaction times for a certain clicking task follows an approximately normal distribution with a mean of 284 milliseconds and a standard deviation of 50 milliseconds.
a. what percent of people take longer that 300 milliseconds to complete the task?
b. it has been claimed that to be a serious video gamer, a person needs to be in the fastest 1% of all reaction times. what time would a person need to get in this clicking task in order to be considered a serious video gamer?

Explanation:

Step1: Calculate z-score for 300 ms

$z = \frac{X - \mu}{\sigma} = \frac{300 - 284}{50} = 0.32$

Step2: Find upper tail probability

For $z=0.32$, the cumulative probability is 0.6255. Upper tail: $1 - 0.6255 = 0.3745$

Step3: Convert to percentage

$0.3745 \times 100 = 37.45\%$

Step4: Find z for fastest 1%

Fastest 1% corresponds to $z$-score for 1st percentile: $z = -2.33$

Step5: Calculate reaction time for z=-2.33

$X = \mu + z\sigma = 284 + (-2.33)(50) = 284 - 116.5 = 167.5$

Answer:

a. 37.45%
b. 167.5 milliseconds