$Normal distribution & Probability$

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$Normal distribution & Probability$

The grades of a group of 5000 students in an exam are normally distributed with a mean of 75 and a standard deviation of 5. A student from this group is selected randomly.

(i) Find the probability that his/her grade is greater than 90.
(ii) Find the probability that his/her grade is less than 50.
(iii) Find the probability that his/her grade is between 50 and 75.
(iv) Approximately, how many students have grades greater than 75.

Statistics Probability

Emmadavis

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(i) We need to compute the $z$-score first.
\[z=\frac{90-75}{5}=3.\]
Using standard comulative normal table (https://en.wikipedia.org/wiki/Standard_normal_table) we get \[P(x>90)=1-0.99865=0.00135. \]

(ii) Again $z=\frac{50-75}{5}=-5$. So
\[P(z<-5)=.0000002867865.\]

(iii) $P(-5<z<0)= \frac{1}{2}-.0000002867865=0.49999971321.$

(iv) $z=\frac{75-75}{5}=0$. Since $P(z>0)=\frac{1}{2}$, approximately $\frac{1}{2}\times 5000=2500$ have grades greater than 75.

Erdos

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$Normal distribution & Probability$

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