2. Children’s heights are normally distributed with a mean of 100 cm and a standard deviation of 6 cm. What proportion of children’s heights is smaller than 94?
To find the proportion of children's heights that is smaller than 94 cm, we need to calculate the z-score and then use a standard normal distribution table.
Step 1: Calculate the z-score
The z-score formula is given by:
z = (x - μ) / σ
Where:
x = the value we want to find the proportion for (94 cm)
μ = the mean (100 cm)
σ = the standard deviation (6 cm)
Plugging in the values:
z = (94 - 100) / 6
z = -1
Step 2: Use the standard normal distribution table
The standard normal distribution table provides the area (proportion) to the left of a given z-score.
Looking up the z-score of -1 in the standard normal distribution table, we find that the area/proportion to the left of -1 is 0.1587.
Step 3: Interpret the result
This means that approximately 15.87% of children's heights are smaller than 94 cm.
Therefore, the proportion of children's heights that is smaller than 94 cm is 0.1587 or 15.87%.