Question

A set of middle school student heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. Let X=, the height of a randomly selected student from this set.

Answer 1

Answer:

The proportion of students whose height are lower than Darnell's height is 71.57%

Step-by-step explanation:

The complete question is:

A set of middle school student heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. Darnel is a middle school student with a height of 161.4cm.

What proportion of proportion of students height are lower than Darnell's height.

Answer:

We first calculate the z-score corresponding to Darnell's height using:

$Z=\frac{X-\mu}{\sigma}$

We substitute x=161.4 , $\mu=150$, and $\sigma=20$ to get:

$Z=\frac{161.4-150}{20} \\Z=0.57$

From the normal distribution table, we read 0.5 under 7.

The corresponding area is 0.7157

Therefore the proportion of students whose height are lower than Darnell's height is 71.57%

Answer 2

The value of P(X < 115) of the students height normally distributed is;

P(Z < -1.75) = 0.04

Given that :

Population Mean; μ = 150

Population Standard deviation; σ = 20

We want to find P(X < 115) which was missing in the question

Let us find the z-score first

Formula for z-score is:

z = (x' - μ)/σ

Plugging in the relevant values gives;

z = (115 - 150)/20

z = -35/20

z = -1.75

From online p-value from z-score calculator, the p-value is;

P(Z < -1.75) = 0.04

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