a) feet
b)
c)
d)
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".
a. (2pts) What is the 70th percentile of the distribution of stopping distances? (Show work, give units)
Let X the random variable that represent variable of interest, and for this case we know the distribution for X is given by:
Where and
So we are interested on a value c that satisfy the following condition:
And the best way to solve this problem is using the normal standard distribution and the z score given by:
So we can find a z score in the normal standard distribution that accumulates 0.7 of the area on the left and 0.3 on the right. And this value on this case is z=0.524. And now we can solve X from the z score formula:
feet
b. (2pts) What is the probability that a randomly selected car will have a stopping distance less than 115 feet? (Give the proper probability statements/notation, show work, and give value to 4 decimal places)
On this case we want this probability:
And we can solve this using again the normal standard distribution and the z score given by:
If we apply the z score formula we got:
And we can find this probability on this way:
c. (4pts) What is the probability that a randomly selected sample of 5 cars in the study will have a mean stopping distance of at least 130 feet? (Give the proper probability statements/notation, show work, and give value to 4 decimal places)
Let represent the sample mean, the distribution for the sample mean by the ceentral limit theorem is given by:
On this case
And we want this probability:
And using the z score formula we got this:
And we can use the complement rule like this:
d. (4pts) What is the probability that a randomly selected sample of 15 cars in the study will have a mean stopping distance between 120 and 130 feet? (Give the proper probability statements/notation, show work, and give value to 4 decimal places)
On this case
And we want this probability:
And using the z score formula we got this:
And we can find this probability on this way: