Final answer:
To calculate the probability that a randomly selected woman weighs between 140 lb and 181 lb when the mean weight is 166 lb and the standard deviation is 49 lb, we find the z-scores for 140 lb and 181 lb, then use a z-table or statistical calculator to find the probabilities. The difference between the upper and lower probabilities gives us the desired probability, which is approximately 32.36%.
Step-by-step explanation:
To solve the mathematical problem completely, we need to determine the probability that a randomly selected woman will weigh between 140 lb and 181 lb, given that the weights of women are normally distributed with a mean of 166 lb and a standard deviation of 49 lb.
First, we convert the weights into z-scores using the formula:
Z = (X - μ) / σ
where X is the value from the dataset, μ is the mean, and σ is the standard deviation.
For X = 140 lb (lower bound):
Zlower = (140 - 166) / 49 = -0.53
For X = 181 lb (upper bound):
Zupper = (181 - 166) / 49 = 0.31
Next, we look up the corresponding probabilities for these z-scores in a z-table or use a statistical calculator. The probabilities are:
P(Z < -0.53) = 0.2981
P(Z < 0.31) = 0.6217
Finally, we find the probability that the weight lies between 140 lb and 181 lb by subtracting the lower probability from the upper probability:
P(140 lb < X < 181 lb) = P(Z < 0.31) - P(Z < -0.53) = 0.6217 - 0.2981 = 0.3236
Therefore, the probability that a randomly selected woman will weigh between 140 lb and 181 lb is approximately 0.3236 or 32.36%.