Question

The score on an exam from a certain MAT 112 class, X, is normally distributed with μ=77.7 and σ=9.6.

Answer 1

Answer:

a) $P(X>65.7)$

b) $P(X>65.7)=P(\frac{X-\mu}{\sigma}>\frac{65.7-\mu}{\sigma})=P(Z>\frac{65.7-77.7}{9.6})=P(z>-1.25)$

And we can find this probability using the complement rule:

$P(z>-1.25)=1-P(z$

c) $P(X>86.7)=P(\frac{X-\mu}{\sigma}>\frac{86.7-\mu}{\sigma})=P(Z>\frac{86.7-77.7}{9.6})=P(z>0.9375)$

And we can find this probability using the complement rule:

$P(z>0.9375)=1-P(z$

d) $P(65.7$

And we can find this probability with this difference:

$P(-1.25$

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.  

$P(-1.25$

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the scores for the MAT112 of a population, and for this case we know the distribution for X is given by:

$X \sim N(77.7,9.6)$  

Where $\mu=77.7$ and $\sigma=9.6$

Part a

We can write the event ''a score over 65.7'' like this:

$P(X>65.7)$

Part b

We are interested on this probability

$P(X>65.7)$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we apply this formula to our probability we got this:

$P(X>65.7)=P(\frac{X-\mu}{\sigma}>\frac{65.7-\mu}{\sigma})=P(Z>\frac{65.7-77.7}{9.6})=P(z>-1.25)$

And we can find this probability using the complement rule:

$P(z>-1.25)=1-P(z$

Part

$P(X>86.7)$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we apply this formula to our probability we got this:

$P(X>86.7)=P(\frac{X-\mu}{\sigma}>\frac{86.7-\mu}{\sigma})=P(Z>\frac{86.7-77.7}{9.6})=P(z>0.9375)$

And we can find this probability using the complement rule:

$P(z>0.9375)=1-P(z$

Part d

$P(65.7$

And we can find this probability with this difference:

$P(-1.25$

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.  

$P(-1.25$