The decimal representation of any number is a linear combination of powers of 10. In other words, given a number like 123.456, we can expand it as
for any
, so the above is the same as
Similarly, we can write
Now it's a question of reducing the fraction as much as possible. We have so
Answer:
2.5% of the drivers have a reaction time of more than 1.94 seconds.
16% of the drivers have a reaction time of less than 1.58 seconds
84% of the drivers have a reaction time of less than 1.82 seconds.
Step-by-step explanation:
The 68-95-99.7 rule states that, when X is an observation from a random moundshaped (normally distributed) value with mean and standard deviation
, we have these following probabilities:
There is a 68% probability that X is within 1 standard deviation of the mean(34% probability that is above, 34% probability that is below).
There is a 95% probability that X is within 2 standard deviations of the mean(47.5% above, 47.5% below)
There is a 99.7% probability that X is within 3 standard deviations of the mean(49.85% above, 49.85% below).
In our problem, we have that:
The mean is
The standard deviation is
What percentage of drivers have a reaction time more than 1.94 seconds?
1.94 is two standard deviations above the mean.
There is a 50% probability that X is below the mean and 50% above. If it is above, there is 95% probability that the driver has a reaction time within 2 standard deviations of the mean, this means a reaction time of LESS THAN 1.94 seconds.
So the probability that he has a reaciton time of more than 1.94 seconds is:
2.5% of the drivers have a reaction time of more than 1.94 seconds.
What percentage of drivers have a reaction time less than 1.58 seconds?
1.58 seconds is one standard deviation below the mean
Of those 50% who are below the mean, 68% are within one standard deviation of the mean. This means that 32% percent of those are below one standard deviation of the mean. So
16% of the drivers have a reaction time of less than 1.58 seconds
What percentage of drivers have a reaction time less than 1.82 seconds?
1.58 seconds is one standard deviation above the mean
50% of the drivers are below the mean. So 50% already have a reaction time of less than 1.82 seconds.
Of the 50% that are above the mean, 68% are within one standard deviation. So
84% of the drivers have a reaction time of less than 1.82 seconds.
