Final answer:
The weights that represent the central 78% of the newborn babies at the local hospital are approximately 5.87 lbs to 10.33 lbs.
Step-by-step explanation:
The weights of newborn babies at the local hospital are normally distributed with a mean of 8.1 lbs and a standard deviation of 1.8 lbs. To find the weights that represent the central 78% of the babies, we need to find the z-scores corresponding to the percentiles. The z-scores for the lower and upper percentiles can be found using the z-score formula, z = (x - μ) / σ, where x is the value of interest, μ is the mean, and σ is the standard deviation.
To find the lower percentile, we need to find the z-score that corresponds to the 11th percentile. Using a standard normal distribution table or a calculator, we can find that the z-score is approximately -1.195.
To find the upper percentile, we need to find the z-score that corresponds to the 89th percentile. Using a standard normal distribution table or a calculator, we can find that the z-score is approximately 1.195.
To convert the z-scores back to weights, we can use the formula x = zσ + μ. Substituting in the z-scores and the given mean and standard deviation, we can find the weights as follows:
Lower weight: -1.195 * 1.8 + 8.1 = 5.87 lbs
Upper weight: 1.195 * 1.8 + 8.1 = 10.33 lbs