High School

If the average height for adult men in the United States is approximately normally distributed with a mean of 70 inches and a standard deviation of about 3 inches, it follows from the 68-95-99.7 rule that:

1. About 95 percent of the adult men in the U.S. are between 64 inches and 76 inches tall.
2. 99.7 percent of the men in the U.S. are between 61 inches and 79 inches tall.
3. 68 percent of the men in the U.S. are between 67 inches and 73 inches in height.

Answer :

Let's analyze the statement using the 68-95-99.7 rule, which is a guideline for understanding normal distributions. It states that:

  1. 68% of the data falls within one standard deviation of the mean.
  2. 95% of the data falls within two standard deviations of the mean.
  3. 99.7% of the data falls within three standard deviations of the mean.

Given:

  • Mean height ([tex]\mu[/tex]) = 70 inches
  • Standard deviation ([tex]\sigma[/tex]) = 3 inches

Now, let's examine the statements:

  1. About 95 percent of the adult men in the U.S. are between 73 inches and 76 inches tall.

    • Analysis:
      • Two standard deviations from the mean is from [tex]\mu - 2\sigma[/tex] to [tex]\mu + 2\sigma[/tex]:
      • [tex]70 - 2(3) = 64[/tex] inches to [tex]70 + 2(3) = 76[/tex] inches.
      • This range should be 64 inches to 76 inches, not 73 inches to 76 inches. Thus, this statement is incorrect.

  2. 99.7 percent of the men in the U.S. are between 61 inches and 79 inches tall.

    • Analysis:
      • Three standard deviations from the mean is from [tex]\mu - 3\sigma[/tex] to [tex]\mu + 3\sigma[/tex]:
      • [tex]70 - 3(3) = 61[/tex] inches to [tex]70 + 3(3) = 79[/tex] inches.
      • This range accurately reflects the 99.7% range, so this statement is correct.

  3. 68 percent of the men in the U.S. are between 64 inches and 67 inches in height.

    • Analysis:
      • One standard deviation from the mean is from [tex]\mu - \sigma[/tex] to [tex]\mu + \sigma[/tex]:
      • [tex]70 - 3 = 67[/tex] inches to [tex]70 + 3 = 73[/tex] inches.
      • The correct range should be 67 inches to 73 inches. Therefore, this statement is incorrect.

To summarize, out of the three statements, only the second one is correct. The other two statements do not align with the 68-95-99.7 rule when applied correctly.

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