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Assume that a normal distribution of data has a mean of 24 and a standard deviation of 7. Use the 68-95-99.7 rule to find the percentage of values that lie above 17.

Answer :

To solve this problem, we use the 68-95-99.7 rule, also known as the Empirical Rule, which helps us understand the distribution of data in a normal distribution.

Here's the step-by-step explanation:

  1. Identify the Given Information:

    • Mean ([tex]\mu[/tex]) = 24
    • Standard Deviation ([tex]\sigma[/tex]) = 7
    • We need to find the percentage of values above 17.

  2. Determine How Many Standard Deviations 17 is From the Mean:

    • We calculate the z-score to find how many standard deviations 17 is from the mean:
      [tex]z = \frac{X - \mu}{\sigma}[/tex]
      [tex]z = \frac{17 - 24}{7}[/tex]
      [tex]z = \frac{-7}{7} = -1[/tex]
      This result means 17 is 1 standard deviation below the mean.

  3. Apply the 68-95-99.7 Rule:

    • The 68-95-99.7 rule states:
      • About 68% of data falls within 1 standard deviation of the mean.
      • Therefore, 34% of the data lies between the mean and 1 standard deviation below (since it's symmetric).
    • Since we're interested in the values above 17, we combine the 50% of data above the mean (24) and the 16% of data that is between the mean and one standard deviation below.

  4. Calculate the Percentage Above 17:

    • Percentage of data above the mean (24) is 50%.
    • Percentage of data between 17 and 24 (which is 1 standard deviation below the mean) is 34%.
    • Therefore, the percentage of values above 17:
      [tex]50\% + 34\% = 84\%[/tex]

So, 84% of the values are above 17 in this normal distribution.

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