The lengths of the eggs of a species of bird are roughly normally distributed, with a mean of 32 mm and an SD of 1.2 mm.
1.Approximately what is the 99th percentile of the lengths? Give your answer in mm but please do not enter the units.
2.Approximately 50% of the eggs have lengths in the range 32 mm plus or minus ____________ mm.
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To find the answer to these questions, we need to use the properties of the normal distribution. In this case, we have a normally distributed variable, which is the lengths of the eggs of a bird species.
1. To find the 99th percentile of the lengths, we can use the Z-score formula. The Z-score represents the number of standard deviations a value is from the mean. The formula for calculating the Z-score is:
Z = (X - μ) / σ
Where:
Z is the Z-score,
X is the value we want to find the percentile for,
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.
In this case, we want to find the 99th percentile, which means we need to find the Z-score that corresponds to this percentile. The Z-score for the 99th percentile is approximately 2.33 (you can look this up in a standard normal distribution table).
To find the corresponding value, we can rearrange the formula:
X = Z * σ + μ
Substituting the values we have:
X = 2.33 * 1.2 + 32
Calculating this, we find that the approximately 99th percentile of the lengths is 34.8 mm.
2. To find the range of lengths that includes approximately 50% of the eggs, we can use the concept of the standard deviation. In a normal distribution, about 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and approximately 99.7% falls within three standard deviations.
Since we want to find a range that includes approximately 50% of the eggs, we need to find the range within one standard deviation from the mean.
If we take the mean of 32 mm and add/subtract one standard deviation of 1.2 mm, the range of lengths that includes approximately 50% of the eggs is:
32 mm ± 1.2 mm
This means that approximately 50% of the eggs have lengths in the range of 30.8 mm to 33.2 mm.