The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.4 seconds and a standard deviation of 0.05 seconds.(a) what is the probability that a reaction requires more than 0.5 second? (b) what is the probability that a reaction requires between 0.4 and 0.5 second? (c) what is the reaction time that is exceeded 90% of the time?

a)

b)

c)

So the value of height that separates the bottom 90% of data from the top 10% is 0.464.  

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the reaction time of a driver to visual stimulus of a population, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability using the complement rule:

And using the normal standard table or excel we have this:

Part b

And we can find this probability on this way:

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

Part c

For this part we want to find a value a, such that we satisfy this condition:

  (a)

  (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.90 of the area on the left and 0.10 of the area on the right it's z=1.28. On this case P(Z<1.28)=0.90 and P(z>1.28)=0.1

If we use condition (b) from previous we have this:

But we know which value of z satisfy the previous equation so then we can do this:

And if we solve for a we got

So the value of height that separates the bottom 90% of data from the top 10% is 0.464.  

no this is a nonlinear negative

b