The probability that the sample mean of weights is between 58 and 61 lbs is approximately 0.0857 or 8.57%.
Sample mean distribution: Since we're sampling from a normally distributed population with a large sample size (42), the central limit theorem applies. This means the distribution of the sample means will also be normal, with:
Mean = Population mean = 60 lbs.
Standard deviation = Population SD / sqrt(sample size) = 3 lbs / sqrt(42) = 0.44 lbs.
Finding the desired probability: We want the probability that the sample mean falls between 58 and 61 lbs. This corresponds to a range of one standard deviation (0.44 lbs) on either side of the population mean (60 lbs).
Z-scores and the standard normal table: We can use Z-scores to calculate this probability. Z-scores represent how many standard deviations a specific value is away from the mean. For our desired range, the Z-scores are:
Lower limit: (58 lbs - 60 lbs) / 0.44 lbs = -0.45
Upper limit: (61 lbs - 60 lbs) / 0.44 lbs = 0.23
Consulting the standard normal table: This table gives the probability of a standard normal variable falling below a certain Z-score. We can use it twice:
Find the probability below -0.45 (lower limit): 0.6760
Find the probability below 0.23 (upper limit): 0.5917
Final probability calculation: The probability that the sample mean falls between 58 and 61 lbs is the difference between these two probabilities: 0.5917 - 0.6760 = 0.0857.