The probability of having a height that is between 158 cm and 163 cm is 0.69011 and the value of x is 169 centimeters
The given parameters are:
Calculate the z-scores at x = 158 and x = 163 using:
[tex]z = \frac{x - \mu}{\sigma}[/tex]
So, we have:
[tex]z = \frac{158 - 161}{6} = -0.5[/tex]
[tex]z = \frac{163 - 161}{6} = 0.33[/tex]
The probability is then represented as:
P(158 ≤ x ≤ 161) = P(-0.5 ≤ z ≤ 0.33)
Using the z table, we have:
P(158 ≤ x ≤ 161) = 0.69011
Hence, the probability of having a height that is between 158 cm and 163 cm is 0.69011
The probability is represented as:
P(X ≥ x) = 1%
Express as decimal
P(X ≥ x) = 0.10
The z-score at p = 0.10 is:
z = 1.282
Substitute z = 1.282 in [tex]z = \frac{x - \mu}{\sigma}[/tex]
[tex]1.282 = \frac{x - \mu}{\sigma}[/tex]
Make x the subject
[tex]x = 1.282\sigma + \mu[/tex]
Substitute values for standard deviation and mean
x = 1.282 * 6 + 161
Evaluate
x = 168.692
Approximate
x = 169
Hence, the value of x is 169 centimeters
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