Entry-to-a-certain-University-is-determined-by-a-national-test-The-scores-on-this-test-are-normally-distributed-with-a-mean-of-500-and-a-standard-deviation-of-100-a-Find-the-probability-that-a

Question Number 169802 by MathsFan last updated on 09/May/22

$$ \\ $$Entry to a certain University is determined by a national test. The scores on this test are normally
distributed with a mean of 500 and a standard deviation of 100.
a. Find the probability that a student’s total scores will be
i. Greater than 850
ii. Less than 550
iii. Between 300 and 490

Answered by shikaridwan last updated on 10/May/22

f(x)=(1/( (√(2π))σ))e^(−(((((x−μ)/σ))^2 )/2)) here σ=100 μ=500 P(X>850)=(1/(100(√(2π))))∫_(850 ) ^∞ exp(−(((((x−500)/(100)))^2 )/2))dx =(1/( (√(2π))))∫_(3.5) ^∞ exp(−t^2 /2)dt t=((x−500)/(100)) =(1/( (√(2π))))∫_0 ^∞ exp(−t^2 /2)dt−(1/( (√(2π))))∫_0 ^(3.5) e^(−t^2 /2) dt =(1/2)−(1/( (√π)))∫_0 ^((√2)×3.5) e^(−u^2 ) du =(1/2)(1−erf((√2)×3.5)) P(X<550)=∫_(−∞) ^(550) f(x)dx P(300<X<490)=∫_(300) ^(490) f(x)dx

$${f}\left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}\pi}\sigma}{e}^{−\frac{\left(\frac{{x}−\mu}{\sigma}\right)^{\mathrm{2}} }{\mathrm{2}}} \\ $$$${here}\:\sigma=\mathrm{100}\:\:\:\:\mu=\mathrm{500} \\ $$$${P}\left({X}>\mathrm{850}\right)=\frac{\mathrm{1}}{\mathrm{100}\sqrt{\mathrm{2}\pi}}\int_{\mathrm{850}\:} ^{\infty} {exp}\left(−\frac{\left(\frac{{x}−\mathrm{500}}{\mathrm{100}}\right)^{\mathrm{2}} }{\mathrm{2}}\right){dx} \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}\pi}}\int_{\mathrm{3}.\mathrm{5}} ^{\infty} {exp}\left(−{t}^{\mathrm{2}} /\mathrm{2}\right){dt}\:\:\:\:{t}=\frac{{x}−\mathrm{500}}{\mathrm{100}} \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}\pi}}\int_{\mathrm{0}} ^{\infty} {exp}\left(−{t}^{\mathrm{2}} /\mathrm{2}\right){dt}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}\pi}}\int_{\mathrm{0}} ^{\mathrm{3}.\mathrm{5}} {e}^{−{t}^{\mathrm{2}} /\mathrm{2}} {dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\:\sqrt{\pi}}\int_{\mathrm{0}} ^{\sqrt{\mathrm{2}}×\mathrm{3}.\mathrm{5}} {e}^{−{u}^{\mathrm{2}} } {du} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−{erf}\left(\sqrt{\mathrm{2}}×\mathrm{3}.\mathrm{5}\right)\right) \\ $$$${P}\left({X}<\mathrm{550}\right)=\int_{−\infty} ^{\mathrm{550}} {f}\left({x}\right){dx} \\ $$$${P}\left(\mathrm{300}<{X}<\mathrm{490}\right)=\int_{\mathrm{300}} ^{\mathrm{490}} {f}\left({x}\right){dx} \\ $$

Commented by MathsFan last updated on 10/May/22

$${wow} \\ $$$${thank}\:{you}\:{sir} \\ $$

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Entry-to-a-certain-University-is-determined-by-a-national-test-The-scores-on-this-test-are-normally-distributed-with-a-mean-of-500-and-a-standard-deviation-of-100-a-Find-the-probability-that-a

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