Question

For a normal distribution of raw scores with µ= 75, σ = 8, answer the following.  

What is the probability of p(71 < X < 83) ?     __________

Find the percentile ranking for the raw score   X = 65th   ______ percentile

Answer 1

We are given a random variable with the following distribution:
$\\ X \sim N \left( \mu = 75 \ , \ \sigma^2 = 8^2 \right ) \\ \Rightarrow Z = \frac{X-\mu}{\sigma} = \frac{X-75}{8} \sim N(0,1) \text{, the standard normal distribution.}$

Part 1

The probability that X lies in the interval (71,83) is given by:
$\begin{align*} \boldsymbol{P(71<X<83)} &= P\left(\frac{71-75}{8} < \frac{X-\mu}{\sigma} < \frac{83-75}{8} \right ) \\ &= P(-0.5 <Z<1) \\ &= P(Z<1) - P(Z\le -0.5) \\ &\text{Using the table for standard normal distribution, we get:} \\ &= 0.841345 - 0.308538 \\ &= \bf 0.532807 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [ANSWER] \end{align*}$

Part 2

The percentile ranking for the raw score X = 65 can be found using:

$\begin{align*} P(X<65) &= P\left(\frac{X-\mu}{\sigma} < \frac{65-75}{8} \right ) \\ &= P(Z<-1.25) \\ &= 0.10565 \\ &= 10.565\% \end{align*}$

Thus, the percentile ranking for the raw score X = 65 is 10.565 th percentile. [ANSWER]

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