Questions: In a survey of a group of men, the heights in the 20-29 age group were normally distributed, with a mean of 68.6 inches and a standard deviation of 4.0 inches. A study participant is randomly selected. Complete parts (a) through (d) below. (a) Find the probability that a study participant has a height that is less than 68 inches. The probability that the study participant selected at random is less than 68 inches tall is (Round to four decimal places as needed.)

In a survey of a group of men, the heights in the 20-29 age group were normally distributed, with a mean of 68.6 inches and a standard deviation of 4.0 inches. A study participant is randomly selected. Complete parts (a) through (d) below. (a) Find the probability that a study participant has a height that is less than 68 inches. The probability that the study participant selected at random is less than 68 inches tall is (Round to four decimal places as needed.)
Transcript text: In a survey of a group of men, the heights in the 20-29 age group were normally distributed, with a mean of 68.6 inches and a standard deviation of 4.0 inches. A study participant is randomly selected. Complete parts (a) through (d) below. (a) Find the probability that a study participant has a height that is less than 68 inches. The probability that the study participant selected at random is less than 68 inches tall is $\square$ (Round to four decimal places as needed.)
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Solution

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Solution Steps

Step 1: Calculate the Z-Score

Given:

  • Mean height, \(\mu = 68.6\) inches
  • Standard deviation, \(\sigma = 4.0\) inches
  • Height value, \(X = 68\) inches

The Z-score is calculated using the formula: \[ Z = \frac{X - \mu}{\sigma} \]

Substituting the given values: \[ Z = \frac{68 - 68.6}{4.0} = -0.15 \]

Step 2: Find the Probability Using the Z-Score

Using the cumulative distribution function (CDF) of the standard normal distribution, we find the probability corresponding to \(Z = -0.15\).

\[ P(Z < -0.15) \approx 0.4404 \]

Final Answer

The probability that the study participant selected at random is less than 68 inches tall is: \[ \boxed{0.4404} \]

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