The distribution of heights of women aged 20 to 29 is approximately Normal with mean 63.6 inches and standard deviation 3 inches.
The height (± 0.1 inch) of the middle 68% of young women falls between a low of inches and a high of inches.
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To find the range for the middle 68% of heights, we need to use the concept of a z-score and the properties of the standard normal distribution (a specific type of normal distribution with mean 0 and standard deviation 1). Here are the steps to determine the range:
Step 1: Convert the given measurements (mean and standard deviation) from the given normal distribution to the standard normal distribution by using the z-score formula:
Z = (X - μ) / σ
where:
Z = the z-score
X = the value in question
μ = the mean of the distribution
σ = the standard deviation of the distribution
In this case, we know the mean (μ) is 63.6 inches and the standard deviation (σ) is 3 inches. We need to find the z-scores for the two values of height that define the range.
Step 2: Determine the z-scores corresponding to the percentiles that define the middle 68%. Since the normal distribution is symmetric, we take half of the percentage (68% / 2 = 34%) and find the corresponding z-scores using a standard normal distribution table or a calculator.
Step 3: Calculate the heights for the given z-scores using the formula:
X = Z * σ + μ
where:
X = the value (height) in question
Z = the z-score
σ = the standard deviation of the distribution
μ = the mean of the distribution
Step 4: Round the calculated heights to the nearest 0.1 inch to match the given precision.
Let's calculate the range:
Step 1: Convert the measurements to the standard normal distribution:
Z_low = (low - μ) / σ
Z_high = (high - μ) / σ
Since we want to find the range within the middle 68%, we calculate the z-scores for the percentiles 16% and 84% (half of 68%) because they correspond to a range of ±34% around the mean.
Z_low = -0.995
Z_high = 0.995
Step 3: Calculate the heights for the given z-scores:
X_low = Z_low * σ + μ
X_high = Z_high * σ + μ
X_low = -0.995 * 3 + 63.6 ≈ 59.615 inches
X_high = 0.995 * 3 + 63.6 ≈ 67.585 inches
Step 4: Round the heights to the nearest 0.1 inch:
X_low ≈ 59.6 inches
X_high ≈ 67.6 inches
Therefore, the height (± 0.1 inch) of the middle 68% of young women falls between a low of 59.6 inches and a high of 67.6 inches.