Question

Mrs. G's newborn baby girl sleeps for approximately 20.1 hours per day. Newborn sleep schedules are normally distributed with a standard deviation of 0.8 hours. If you know Mrs. G's daughter is at the 67ᵗʰ percentile for sleep, what is the mean sleep time of all newborn baby girls?

Answer 1

To determine the mean sleep time of all newborn baby girls, given that Mrs. G's daughter sleeps 20.1 hours per day at the 67th percentile, we use the Z-score formula with the given standard deviation of 0.8 hours. The Z-score for the 67th percentile is approximately 0.44. After solving the Z-score equation, we find that the mean sleep time is approximately 19.748 hours.

Step-by-step explanation:

To find the mean sleep time of all newborn baby girls given that Mrs. G's daughter, who sleeps for approximately 20.1 hours per day, is at the 67th percentile, we need to use the properties of the normal distribution. Since the standard deviation is given as 0.8 hours, and we know the percentile position of the daughter's sleep time, we can use the Z-score formula associated with the 67th percentile to calculate the mean.

To start, we look up the Z-score that corresponds to the 67th percentile in the standard normal distribution table, which is approximately 0.44. The Z-score formula is: Z = (X - μ) / σ, where X is a value from the distribution, μ is the mean, and σ is the standard deviation. Substituting the known values: 0.44 = (20.1 - μ) / 0.8. To find the mean (μ), we solve for μ: μ = 20.1 - (0.44 × 0.8). μ = 20.1 - 0.352. μ = 19.748 hours. Therefore, the mean sleep time for all newborn baby girls is approximately 19.748 hours

Answer 2

The mean sleep time for all newborn baby girls is approximately 20.7 hours.

Assuming a normal distribution for newborn sleep schedules, we can calculate the mean sleep time for all newborn baby girls using the information provided about Mrs. G's daughter. Here's how:

Standardize the percentile: The 67th percentile corresponds to a z-score of approximately 0.44. This is because 67% of the population falls below this point in a standard normal distribution.

Calculate the mean using the z-score: We know the sleep time of Mrs. G's daughter (20.1 hours) and the standard deviation (0.8 hours). We can use the z-score to relate these values to the mean:

mean = daughter's sleep time + z-score * standard deviation

mean = 20.1 hours + 0.44 * 0.8 hours

mean ≈ 20.7 hours