High School

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Let [tex]x[/tex] be normally distributed with a mean of 6 and a variance of 25.

a. Sketch the density curve of [tex]x[/tex] using the 68-95-99.7 Rule.

Answer :

To sketch the density curve of x, normally distributed with a mean of 6 and a variance of 25, one needs to draw a bell-shaped curve, label the mean at the center, and mark standard deviations away from the mean, shading areas that represent the percentages according to the 68-95-99.7 Rule.

The student is asked to sketch the density curve of a normally distributed variable x with a mean (\u03bc) of 6 and a variance (\\(\sigma^2\)) of 25. The standard deviation (\(\sigma\)) is the square root of the variance, which is 5 in this case. According to the 68-95-99.7 Rule (also known as the Empirical Rule), approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

To sketch the curve:

  • Draw a bell-shaped curve.
  • Label the mean (6) at the center of the curve.
  • Mark one standard deviation away from the mean on both sides (1\(\sigma\) = 1 and 11).
  • Label two standard deviations away from the mean (2\(\sigma\) = -4 and 16).
  • Label three standard deviations away from the mean (3\(\sigma\) = -9 and 21).
  • Shade the respective areas under the curve to represent the percentages (68%, 95%, and 99.7%).

The resulting graph would illustrate these intervals showing the distribution of x around the mean, representing the spread and concentration of the data.

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