The lifespans of gorillas in a particular zoo are normally distributed. The average gorilla lives
\[20.8\] years; the standard deviation is
\[3.1\] years.
Use the empirical rule
\[(68-95-99.7\%)\] to estimate the probability of a gorilla living between
\[11.5\] and
\[27\] years.
84
\[\%\]
Hint #11 / 8
A normal distribution curve is plotted along a horizontal axis. The axis is marked at 20.8. A vertical line segment rises from 20.8 to the center of the curve. The segment is shaded pink.
\[20.8\]
We know the lifespans are normally distributed with an average lifespan of
\[20.8\] years.
Hint #22 / 8
A normal distribution curve is plotted along a horizontal axis. The axis is marked at 20.8. A vertical line segment rises from 20.8 to the center of the curve. 2 additional vertical segments, each shaded pink, rise from points on the axis marked as 17.7 and 23.9 to the curve.
\[20.8\]
\[17.7\]
\[23.9\]
We know the standard deviation is
\[3.1\] years, so one standard deviation below the mean is
\[17.7\] years and one standard deviation above the mean is
\[23.9\] years.
Hint #33 / 8
A normal distribution curve is plotted along a horizontal axis. The axis is marked at 20.8. A vertical line segment rises from 20.8 to the center of the curve. 4 additional vertical segments rise from points on the axis marked as 14.6, 17.7, 23.9 and 27, to the curve. The segments at 14.6 and 27 are shaded pink.
\[20.8\]
\[17.7\]
\[23.9\]
\[14.6\]
\[27\]
Two standard deviations below the mean is
\[14.6\] years and two standard deviations above the mean is
\[27\] years.
Hint #44 / 8
A normal distribution curve is plotted along a horizontal axis. The axis is marked at 20.8. A vertical line segment rises from 20.8 to the center of the curve. 6 additional vertical segments rise from points on the axis marked as 11.5, 14.6, 17.7, 23.9, 27, and 30.1 to the curve. The segments at 11.5 and 30.1 are shaded pink.
\[20.8\]
\[17.7\]
\[23.9\]
\[14.6\]
\[27\]
\[11.5\]
\[30.1\]
Three standard deviations below the mean is
\[11.5\] years and three standard deviations above the mean is
\[30.1\] years.
Hint #55 / 8
A normal distribution curve is plotted along a horizontal axis. The axis is marked at 20.8. A vertical line segment rises from 20.8 to the center of the curve. 6 additional vertical segments rise from points on the axis marked as 11.5, 14.6, 17.7, 23.9, 27, and 30.1 to the curve. The segments at 11.5 and 27 are shaded pink. A horizontal line segment beneath the axis begins at a solid circle at 11.5 and ends at a solid circle at 27.
\[20.8\]
\[17.7\]
\[23.9\]
\[14.6\]
\[27\]
\[11.5\]
\[30.1\]
We are interested in the probability of a gorilla living between
\[11.5\] and
\[27\] years.
Hint #66 / 8
A normal distribution curve is plotted along a horizontal axis. The axis is marked at 20.8. A vertical line segment rises from 20.8 to the center of the curve. 6 additional vertical segments rise from points on the axis marked as 11.5, 14.6, 17.7, 23.9, 27, and 30.1 to the curve. The segments at 11.5 and 27 are shaded pink. A horizontal line segment beneath the axis begins at a solid circle at 11.5 and ends at a solid circle at 27. The area beneath the curve, but above the axis, between 11.5 and 30.1, is shaded. An arrow indicates the amount of data between 11.5 and 30.1 is 99.7%.
\[20.8\]
\[17.7\]
\[23.9\]
\[14.6\]
\[27\]
\[11.5\]
\[30.1\]
\[99.7\%\]
The empirical rule (or the
\[68-95-99.7\%\] rule) tells us that
\[\green{99.7\%}\] of the gorillas will have lifespans within
\[3\] standard deviations of the average lifespan.
Hint #77 / 8
A normal distribution curve is plotted along a horizontal axis. The axis is marked at 20.8. A vertical line segment rises from 20.8 to the center of the curve. 6 additional vertical segments rise from points on the axis marked as 11.5, 14.6, 17.7, 23.9, 27, and 30.1 to the curve. The segments at 11.5 and 27 are shaded pink. A horizontal line segment beneath the axis begins at a solid circle at 11.5 and ends at a solid circle at 27. The area beneath the curve, but above the axis, between 11.5 and 30.1, is shaded. An arrow indicates the amount of data between 11.5 and 30.1 is 99.7%. Additional arrows indicate that the amount of data between 14.6 and 27 is 95%, and the amount of data between 11.5 and 14.6 and between 27 and 30.1 is 2.35%.
\[20.8\]
\[17.7\]
\[23.9\]
\[14.6\]
\[27\]
\[11.5\]
\[30.1\]
\[99.7\%\]
\[95\%\]
\[2.35\%\]
\[2.35\%\]
It also tells us that
\[\green{95\%}\] of the gorillas will have lifespans within
\[2\] standard deviations of the mean. That leaves
\[99.7\% - 95\% = 4.7\%\] of gorillas between
\[2\] and
\[3\] standard deviations of the mean, or
\[\red{2.35\%}\] on either side of the distribution.
Hint #88 / 8
The probability of a particular gorilla living between
\[11.5\] and
\[27\] years is
\[\red{2.35\%} + \green{95\%}\], or
\[97.35\%\].
