Suppose that weights are bell shaped with a mean of 120 pounds and a standard deviation of 10 pounds. What percentage of weights is between 110 and 130 pounds
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion between the two Z scores.
To determine the percentage of weights between 110 and 130 pounds, we will use the concept of the standard normal distribution.
Step 1: Convert the given weights into z-scores.
The formula for calculating the z-score is:
z = (x - μ) / σ
where z is the z-score, x is the given weight, μ is the mean, and σ is the standard deviation.
For the lower weight of 110 pounds:
z1 = (110 - 120) / 10 = -1
For the higher weight of 130 pounds:
z2 = (130 - 120) / 10 = 1
Step 2: Find the corresponding area under the standard normal distribution curve.
Using a standard normal distribution table or a statistical calculator, we can find the cumulative probability of the z-scores.
For z1 = -1, the cumulative probability is approximately 0.1587 (from the table or calculator).
For z2 = 1, the cumulative probability is approximately 0.8413 (from the table or calculator).
Step 3: Calculate the percentage within the range.
To determine the percentage of weights between 110 and 130 pounds, subtract the lower cumulative probability from the higher cumulative probability and convert it to a percentage.
Percentage = (0.8413 - 0.1587) * 100% ≈ 68%
Therefore, approximately 68% of weights will fall between 110 and 130 pounds.