Assume that the weights of men are normally distributed with a mean of 182.9 pounds and a standard deviation of 40.8 pounds. What proportion of men weigh between 180 and 200 pounds? Round to 4 decimal places.
180 = µ - 0.071 σ
200 = µ + 0.419 σ
So what does your Z table say?
To find the proportion of men who weigh between 180 and 200 pounds, we can use the Z-score formula and the standard normal distribution table.
The Z-score formula is given by Z = (X - μ) / σ, where X is the desired weight, μ is the mean, and σ is the standard deviation.
For X = 180 pounds:
Z1 = (180 - 182.9) / 40.8
For X = 200 pounds:
Z2 = (200 - 182.9) / 40.8
Let's calculate these values:
Z1 = (180 - 182.9) / 40.8
= -0.070588
Z2 = (200 - 182.9) / 40.8
= 0.421569
Now, we need to find the probabilities associated with these Z-scores using the standard normal distribution table.
For Z1 = -0.070588, the corresponding probability is 0.4726.
For Z2 = 0.421569, the corresponding probability is 0.6636.
To find the proportion of men who weigh between 180 and 200 pounds, we subtract the probability corresponding to Z1 from the probability corresponding to Z2:
Proportion = P(180 ≤ X ≤ 200)
= P(Z1 ≤ Z ≤ Z2)
= P(Z ≤ Z2) - P(Z ≤ Z1)
= 0.6636 - 0.4726
= 0.1910
Therefore, approximately 0.1910, or 19.10% of men weigh between 180 and 200 pounds.
To find the proportion of men who weigh between 180 and 200 pounds, we need to calculate the z-scores for both weights and then use the standard normal distribution table.
Step 1: Calculate the z-score for the lower boundary (180 pounds):
Z = (X - μ) / σ
Z = (180 - 182.9) / 40.8
Z = -0.070
Step 2: Calculate the z-score for the upper boundary (200 pounds):
Z = (200 - 182.9) / 40.8
Z = 0.419
Step 3: Use the standard normal distribution table to find the proportion between these z-scores.
Looking up the z-scores in the table, we can find that the cumulative probability or proportion for the lower z-score (-0.070) is 0.4732, and for the upper z-score (0.419) is 0.6631.
Step 4: Subtract the lower cumulative probability from the upper cumulative probability to find the proportion between the two scores:
Proportion = 0.6631 - 0.4732
Proportion = 0.1899
Therefore, approximately 0.1899 or 18.99% of men weigh between 180 and 200 pounds.