Question

The average grade/mark of all college students is 70 with standard deviation of 20. If a random sample of 49 students is taken, what is the probability that the sample average grade/mark is
a) At least 65
b) Less than 78
c) Between 65 and 72

Answer 1

$Given\;that\, \bar x=70,s=20,n=49\\ (a) P(\bar x>65)=P(Z>\frac{65-70}{\frac{20}{\sqrt{49}}})\\ =P(Z>-1.75)=0.5+P(0<Z<1.75)\\ =0.5+0.4599=0.9599\\ (b) P(\bar x<78)=P(Z<\frac{78-70}{\frac{20}{\sqrt{49}}})\\ =P(Z<2.8)=0.5+P(0<Z<2.8)\\ =0.5+0.4974=0.9974\\ (c)P(65<\bar x<72)=P(\frac{65-70}{\frac{20}{\sqrt{49}}})<Z<\frac{72-70}{\frac{20}{\sqrt{49}}})\\ =P(-1.75<Z<0.7)\\ =P(0<Z<1.75)+P(0<Z<0.7)\\ =0.4599+0.2580=0.7179\\$