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Misha Larkins [42]
3 years ago
8

In a recent year, the ACT scores for the math portion of the test were normally distributed, with a mean of 21.1 and a standard

deviation of 5.3. Find the probability that a randomly selected high school student who took the math portion of the ACT has a score that is (a) less than 16, (b) between 19 and 24, and (c) more than 26, and (d) identify any unusual events. Explain your reasoning
Mathematics
1 answer:
Natali5045456 [20]3 years ago
6 0

Answer:

a) P(X

And we can find this probability using the normal standard table or excel:

P(z

b) P(19

And we can find this probability with this difference:

P(-0.396

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

P(-0.396

c) P(X>26)=P(\frac{X-\mu}{\sigma}>\frac{26-\mu}{\sigma})=P(Z>\frac{26-21.1}{5.3})=P(z>0.925)

And we can find this probability using the complement rule and the normal standard table or excel:

P(z>0.925)=1- P(Z

d) We can consider unusual events values above or below 2 deviations from the mean

Lower = \mu -2*\sigma = 21.1 -2*5.3 =10.5

A value below 10.5 can be consider as unusual

Upper = \mu +2*\sigma = 21.1 +2*5.3 =31.7

A value abovr 31.7 can be consider as unusual

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".  

Part a

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

X \sim N(21.1,5.3)  

Where \mu=21.1 and \sigma=5.3

We are interested on this probability

P(X

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

And we can find this probability using the normal standard table or excel:

P(z

Part b

P(19

And we can find this probability with this difference:

P(-0.396

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

P(-0.396

Part c

P(X>26)=P(\frac{X-\mu}{\sigma}>\frac{26-\mu}{\sigma})=P(Z>\frac{26-21.1}{5.3})=P(z>0.925)

And we can find this probability using the complement rule and the normal standard table or excel:

P(z>0.925)=1- P(Z

Part d

We can consider unusual events values above or below 2 deviations from the mean

Lower = \mu -2*\sigma = 21.1 -2*5.3 =10.5

A value below 10.5 can be consider as unusual

Upper = \mu +2*\sigma = 21.1 +2*5.3 =31.7

A value abovr 31.7 can be consider as unusual

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