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valentina_108 [34]
3 years ago
6

Assume that SAT scores are normally distributed with mean mu equals 1518 and standard deviation sigma equals 325. If 1 SAT score

is randomly selected, find the probability that it is greater than 1600. If 81 SAT scores are randomly selected, find the probability that they have a mean greater than 1600.
Mathematics
1 answer:
aleksandr82 [10.1K]3 years ago
4 0

Answer:

P(X>1600)=P(\frac{X-\mu}{\sigma}>\frac{1600-\mu}{\sigma})=P(Z>\frac{1600-1518}{325})=P(z>0.252)

And we can find this probability using the z score formula and the complement rule and we got:

P(z>0.252)=1-P(z

z =\frac{1600-1518}{\frac{325}{\sqrt{81}}}= 2.27

And we can find this probability using the z score formula and the complement rule and we got:

P(z>2.27)=1-P(z

Step-by-step explanation:

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

X \sim N(1518,325)  

Where \mu=1518 and \sigma=325

We want to find this probability:

P(X>1600)

And we can use the z score formula given by:

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

Using this formula we got:

P(X>1600)=P(\frac{X-\mu}{\sigma}>\frac{1600-\mu}{\sigma})=P(Z>\frac{1600-1518}{325})=P(z>0.252)

And we can find this probability using the z score formula and the complement rule and we got:

P(z>0.252)=1-P(z

For the other part we need to take in count that the distribution for the sampel mean if the sample size is large (n>30) is given by:

\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})

And we can use the z score formula given by:

z=\frac{x-\mu}{\frac{sigma}{\sqrt{n}}}

And replacing we got:

z =\frac{1600-1518}{\frac{325}{\sqrt{81}}}= 2.27

And we can find this probability using the z score formula and the complement rule and we got:

P(z>2.27)=1-P(z

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