Final answer:
To find the probability that the mean overhead reach of 50 randomly selected adult females is between 198.0 cm and 200.0 cm, calculate the standard error, convert the reach distances to z-scores, and use the z-scores to determine the probability from a standard normal distribution table.
Step-by-step explanation:
To calculate the probability that the mean overhead reach of a sample of 50 adult females is between 198.0 cm and 200.0 cm, we will use the Central Limit Theorem which tells us that the sampling distribution of the sample mean is normally distributed when the sample size is large (n>30). Given the population mean (μ) is 205.5 cm and the standard deviation (σ) is 8.6 cm, we first need to find the standard error (SE) of the sample mean for a sample size (n) of 50:
SE = σ / √n = 8.6 / √50 ≈ 1.216
Next, we convert the given range into z-scores. The z-score formula is:
Z = (X - μ) / SE
For X = 198.0 cm:
Z1 = (198.0 - 205.5) / 1.216 ≈ -6.17
For X = 200.0 cm:
Z2 = (200.0 - 205.5) / 1.216 ≈ -4.52
We then look up these z-scores in a standard normal distribution table or use a calculator to find the probabilities and subtract the smaller probability from the larger one to find the probability of the sample mean falling between 198.0 cm and 200.0 cm.
Assuming this calculation returns values P1 and P2 for Z1 and Z2, respectively, the probability we are looking for is P2 - P1.
Without the actual z-table or a calculator, we can not provide the exact probability, but the method described will give the answer when these tools are used.