In the standard normal z-distribution, is the area between z =-1 and z between z
ID: 3273921 • Letter: I
Question
In the standard normal z-distribution, is the area between z =-1 and z between z 8) 1 the same as the area -0.5 and z = +1.5? why? (Hint: Draw the curve to answer the question). The lifetime of tires is normally distributed with a mean of 50,000 miles and a standard deviation cf 3,000 miles. The warranty is for 46,000 miles. What proportion of the tires will fail after the warranty? 9) 10) If X is a normal random variable with a mean of 20 and a standard deviation of 5, calculate the following probabilities? 1) Probability that X s 12Explanation / Answer
9) Given that lifetime of tire.
X~ N(mean= 50,000 miles, sd = 3,000 miles)
Also given that the warrenty is for 46,000 miles
Now we have to find P(X > 46,000)
Now convert x = 46,000 into z-score.
z-score is defined as,
z = (x - mean) / sd
z = (46000 - 50000) / 3000 = -1.33
So we have to find P(Z > -1.33)
We can find this probability in EXCEL.
syntax :
=1 - NORMSDIST(z)
where z is z-score.
P(Z > -1.33) = 0.9082
10) Given that,
X ~ N(mean = 20, sd = 5)
Calculate the following probabilities.
1) P(X <=12) :
z-score for x = 12 is,
z = (12 - 20) / 5 = -1.6
That is we have to find P(Z < -1.6).
This probability we can find by using EXCEL.
syntax :
=NORMSDIST(z)
where z is z-score
P(Z < -1.6) = 0.0548
2) P(X >8).
z-score for x = 8 is,
z = (8 - 20) / 5 = -2.4
Now we have to find P(Z > -2.4)
P(Z>-2.4) = 0.9918
3) P(2 <= X <=14)
z-scores for x = 2 and x = 14 are,
z = (2 - 20) / 5 = -3.6
z = (14 - 20)/5 = -1.2
Now we have to find P(-3.6 < Z < -1.2).
P(-3.6 < Z < -1.2) = P(Z < -1.2) - P(Z< -3.6)
= 0.1151 - 0.0002
= 0.1149
11. Given that X ~ N(mean = 50, sd= 8)
Here we have given probability and from this information we have to find x.
P(X > x) = 0.7824
1 - P(X < x) = 0.7824
P(X < x) = 1 - 0.7824
P(X<x) = 0.2176
Now we can find x by using formula,
x = mean + z*sd
where z is z-score for probabiity 0.2176.
z we can find by using EXCEL.
syntax :
=NORMSINV(probability)
where probability = 0.2176
z = -0.78
x = 50 + (-0.78)*8 = 43.76