Answer to Question #264417 in Statistics and Probability for john

Question

Problem:  The mean score on Statistics and Probability test of STEM students is 30.  The standard deviation is 5.  If the teacher administers the test to a class of 40 students, find the probability that the mean of the sample will be between 27 and 31.  Assume the variable is normally distributed.

Step 1:  Draw the properly labelled normal curve

Answer:_________________________________________

Step 2:  Compute the value of the standard deviation of the sample mean

Answer:___________________________________________

Step 3:  Compute the z1 value when x=27 and z2 value when x=31.

Answer:_____________________________________________________

Step 4:  Find the probability that the mean of the sample will be between 27 and 31

Answer:_________________________________________________

Answer 1

1.

probability density for $x=\mu$ :

$f(\mu)=\frac{1}{\sigma\sqrt{2\pi}}$

$f(30)=\frac{1}{5\sqrt{2\pi}}=0.08$

2.

standard deviation of the sample mean:

$\sigma/\sqrt n=5/\sqrt{40}=0.79$

3.

$z_1=\frac{27-30}{0.79}=-3.79$

$z_2=\frac{31-30}{0.79}=1.27$

4.

$P(27<\mu <31)=P(\mu <31)-P(\mu <27)=P(z<1.27)-P(z<-3.79)=$

$=0.8980-0=0.898$