Answer :
By calculating the z-score we found that about 4.95% of boys at the age of 10 are less than 46.5 inches tall.
Explanation:
To solve this problem, we need to use the concept of the standard normal distribution and z-scores. A z-score measures the number of standard deviations a given data point is away from the mean of the distribution. We'll need to calculate the z-score for the value 46.5 and then find the corresponding area under the standard normal curve to the left of that z-score.
The formula for calculating the z-score is:
z = (x - μ) / σ
where:
- x is the value we're interested in (46.5 inches in this case),
- μ is the mean of the distribution (55.9 inches),
- σ is the standard deviation of the distribution (5.7 inches).
Plugging in the values:
z = (46.5 - 55.9) / 5.7
z ≈ -1.6491
Now that we have the z-score, we need to find the area under the standard normal curve to the left of z. This area represents the cumulative probability up to the value of 46.5.
You can use a standard normal distribution table or a calculator to find this area. Alternatively, you can use statistical software like Python or R to perform the calculations.
The area to the left of z ≈ -1.6491 is approximately 0.0495.
This means that about 4.95% of 10-year-old males have a height less than 46.5 inches.
In terms of the steps for using a standard normal distribution table:
- Look up the z-score of -1.64 (or the closest value) in the table.
- The corresponding area is approximately 0.0495.
The area of 0.0495 represents the proportion of 10-year-old males with heights less than 46.5 inches. In other words, about 4.95% of 10-year-old males have a height that is below 46.5 inches.
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