The hieght,X ,of young American women is distributed normal with mean 65.5 and standard deviation 2.5 inches .find the probability of each the following events.
a- X<67
b-64<X<67
1. find
Z-score = (X-mean)/stdev
=(67-65.5)/(2.5)
=(1.5)/2.5
=0.6
2. Look up standard normal distribution tables for one-tail where Z-score = 0.6
P(x<X=67)=0.7257469 [from tables for Z=0.6]
For part b, find similarly P(x<X=64)
then
P(64<x<67)
=P(x<67)-P(x<64)
To find the probability of each event, we can use the standard normal distribution table or the z-score formula. Here's how you can calculate the probabilities:
a) P(X < 67):
To calculate this probability, we need to find the area under the normal curve to the left of 67.
Step 1: Calculate the z-score for the value 67 using the formula:
z = (X - mean) / standard deviation
z = (67 - 65.5) / 2.5
z = 1.5 / 2.5
z = 0.6
Step 2: Look up the z-score in the standard normal distribution table or use a calculator to find the corresponding probability. In this case, the probability is the area to the left of the z-score.
Using the standard normal distribution table, the probability corresponding to a z-score of 0.6 is approximately 0.7257.
So, P(X < 67) is approximately 0.7257.
b) P(64 < X < 67):
To calculate this probability, we need to find the area under the normal curve between 64 and 67.
Step 1: Calculate the z-scores for the two values using the formula:
z1 = (X1 - mean) / standard deviation
z2 = (X2 - mean) / standard deviation
For X = 64:
z1 = (64 - 65.5) / 2.5
z1 = -1.5 / 2.5
z1 = -0.6
For X = 67:
z2 = (67 - 65.5) / 2.5
z2 = 1.5 / 2.5
z2 = 0.6
Step 2: Calculate the probability for each z-score separately.
Using the standard normal distribution table, the probability corresponding to a z-score of -0.6 is approximately 0.2743, and the probability corresponding to a z-score of 0.6 is approximately 0.7257.
Step 3: Subtract the probability corresponding to the lower z-score from the probability corresponding to the higher z-score.
P(64 < X < 67) = P(X < 67) - P(X < 64)
P(64 < X < 67) = 0.7257 - 0.2743
P(64 < X < 67) = 0.4514
So, P(64 < X < 67) is approximately 0.4514.
To find the probability of each event, we need to calculate the area under the normal curve.
a) X < 67:
To find this probability, we need to calculate the area under the curve to the left of 67. We can use the formula for the standard normal distribution (Z-score) to calculate this probability:
Z = (X - μ) / σ
Where:
X = 67 (the value we want to find the probability for)
μ = 65.5 (mean)
σ = 2.5 (standard deviation)
Now we can calculate the Z-score:
Z = (67 - 65.5) / 2.5
Z ≈ 0.6
Next, we look up the area to the left of a Z-score of 0.6 in the standard normal distribution table (also known as the Z-table). The Z-table tells us that the area to the left of 0.6 is approximately 0.7257.
Therefore, the probability of X < 67 is approximately 0.7257.
b) 64 < X < 67:
To find this probability, we need to calculate the area under the curve between 64 and 67. We can use the same method as above, but calculate two Z-scores:
Z1 = (64 - 65.5) / 2.5 ≈ -0.6
Z2 = (67 - 65.5) / 2.5 ≈ 0.6
Next, we look up the area to the left of each Z-score in the Z-table:
For Z1 ≈ -0.6, the area to the left is approximately 0.2743.
For Z2 ≈ 0.6, the area to the left is approximately 0.7257.
To find the area between these two Z-scores, we subtract the smaller area from the larger one:
Probability = 0.7257 - 0.2743
Probability ≈ 0.4514
Therefore, the probability of 64 < X < 67 is approximately 0.4514.