Final answer:
The probability that an adult female randomly selected has a height between 60.5 and 64.5 inches is approximately 0.3313, or four decimal places, 0.3313, using the normal distribution with the given mean and standard deviation.
Step-by-step explanation:
To find the probability that an adult female selected at random will have a height between 60.5 and 64.5 inches, given that the average height of women is normally distributed with a mean (μ) of 65.4 inches and a standard deviation (σ) of 2.31 inches, we will use the standard normal distribution or Z-score formula:
Z = (X - μ) / σ
First, we calculate the Z-scores for both 60.5 inches and 64.5 inches:
- Z for 60.5 inches: Z = (60.5 - 65.4) / 2.31 ≈ -2.12
- Z for 64.5 inches: Z = (64.5 - 65.4) / 2.31 ≈ -0.39
Next, we look up these Z-scores in a standard normal distribution table (or use a calculator that provides probabilities for Z-scores) to find the area under the curve between these two Z-scores. The probability for Z = -2.12 will be around 0.0170 and for Z = -0.39 will be around 0.3483. The probability that a woman's height is between 60.5 and 64.5 inches is the difference between these two values.
Probability = P(Z < -0.39) - P(Z < -2.12) ≈ 0.3483 - 0.0170 = 0.3313
Therefore, the probability is approximately 0.3313 or rounded to four decimal places, 0.3313.