Answer:
a)
b)
And we can find this probability with the complement rule and with the normal standard table or excel and we got:
c)
And we can find this probability with this difference and with the normal standard table or excel and we got:
d) For this case we can use the definition of z score, since we need 85% of the values in the middle of the distribution on the tails we need the remaining and the value for and if we look that accumulates 0.075 of the area on the left of the normal standard distribution we got:
And using the z score formula we got:
And solving for X we got :
And since the distribution is symmetrical we got:
And solving for X we got :
then we have 85% of the values between (10.8; 25.2)
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 duration of a particular type of criminal of a population, and for this case we know the distribution for X is given by:
Where and
Part b
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 with the complement rule and with the normal standard table or excel and we got:
Part c
And we can find this probability with this difference and with the normal standard table or excel and we got:
Part d
For this case we can use the definition of z score, since we need 85% of the values in the middle of the distribution on the tails we need the remaining and the value for and if we look that accumulates 0.075 of the area on the left of the normal standard distribution we got:
And using the z score formula we got:
And solving for X we got :
And since the distribution is symmetrical we got:
And solving for X we got :
then we have 85% of the values between (10.8; 25.2)