Final answer:
Approximately 61.41% of 10-year-old boys are less than 140 cm in height, calculated by finding the z-score for 140 cm given the normal distribution mean of 138 cm and standard deviation of 7 cm.
Step-by-step explanation:
To determine what proportion of 10-year-old boys are less than 140 cm in height, given a normal distribution with a mean of 138 cm and a standard deviation of 7 cm, we first need to calculate the z-score corresponding to 140 cm.
The z-score is calculated using the formula z = (x - μ) / σ, where x is the height of interest, μ is the mean, and σ is the standard deviation. Plugging the values in:
z = (140 cm - 138 cm) / 7 cm = 2 / 7 ≈ 0.29
Next, we look up the z-score in the standard normal distribution table, or use a calculator with normal distribution functions to find the area to the left of z = 0.29, which corresponds to the proportion of boys less than 140 cm tall.
The table or calculator gives us a proportion of about 0.6141. Therefore, approximately 61.41% of 10-year-old boys are less than 140 cm in height.