Final answer:
The probability that the mean weight of 36 randomly selected women is between 140 lb and 211 lb is approximately 73.24%, calculated using the properties of the normal distribution of sample means.
Step-by-step explanation:
To determine the probability that the mean weight of 36 randomly selected women is between 140 lb and 211 lb, we can use the normal distribution properties since we're dealing with a sample mean. The sample mean weight has a normal distribution with a mean (μ) equal to the population mean, which is 143 lb, and a standard deviation (σ) equal to the population standard deviation divided by the square root of the sample size (n), which is 29 lb/√36 = 29 lb/6 = 4.83 lb.
To find the probability of the sample mean falling between 140 lb and 211 lb, we will use the z-score formula: z = (X - μ) / (σ/√ n), where X is the desired sample mean. For X=140 lb, the z-score is z1 = (140 - 143) / 4.83 = -0.62. For X=211 lb, the z-score is z2 = (211 - 143) / 4.83 = 14.08. We use a standard normal distribution table or a calculator to find the probabilities corresponding to these z-scores.
Because z2 corresponds to a very high weight far above the mean and beyond the typical range of the standard normal distribution table, we can consider the probability associated with it to be essentially 1. Therefore, we're mainly interested in the probability for z1. If we look up -0.62 in the z-table, we find that it corresponds to a lower tail probability of approximately 0.2676. Therefore, the probability that the mean weight is above 140 lb is 1 - 0.2676 = 0.7324.
Thus, the probability that the mean weight of 36 randomly selected women falls between 140 lb and 211 lb is approximately 0.7324 or 73.24%.