1)Let X have a normal distribution with mean 6 and standard deviation 3. a) What is the probability that X > 8.7
2)What is the probability that X<0?
3)What is the probability that |X-6| > 1.9 ?
I have got so far:
P(Z > 0.9) but I do not know where to go
from here!
Thank You
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to your Z score(s).
0-0.1
To solve these probability questions involving a normal distribution, you can use the standard normal distribution table or a statistical calculator.
1) To find the probability that X is greater than 8.7, you need to standardize the value using the z-score formula:
z = (X - mean) / standard deviation
In this case, X = 8.7, mean = 6, and standard deviation = 3. Plugging in these values:
z = (8.7 - 6) / 3 = 0.9
Now, you need to find the probability that Z is greater than 0.9 using the standard normal distribution table or a calculator. The table typically provides the area to the left of a given z-score, so you need to find the area to the left of -0.9 and subtract it from 1 (the total area):
P(Z > 0.9) = 1 - P(Z < 0.9)
Look up the z-score 0.9 in the standard normal distribution table and find the corresponding probability (usually denoted as "P(Z < 0.9)"). Subtract this probability from 1 to get the desired result.
2) To find the probability that X is less than 0, you can use the same approach:
z = (0 - 6) / 3 = -2
P(X < 0) is equivalent to P(Z < -2). Look up the z-score -2 in the standard normal distribution table and find the corresponding probability.
3) To find the probability that |X - 6| > 1.9, you can break it down into two separate inequalities:
a) X - 6 > 1.9
b) X - 6 < -1.9
Solving equation (a):
X > 1.9 + 6
X > 7.9
Solving equation (b):
X < -1.9 + 6
X < 4.1
Thus, you are interested in the probability that X is greater than 7.9 or less than 4.1. You can find these probabilities separately and then add them together. Standardize both values using the z-score formula and find the corresponding probabilities using the standard normal distribution table or a calculator.
I hope this helps! Let me know if you have any further questions.