The heights of 1000 students are approximately normally distributed with a mean

Question

The heights of 1000 students are approximately normally distributed with a mean of 174.5 cm and a standard deviation of 6.9 cm. Suppose 200 random samples of size 25 are drawn from this population and the means recorded to the nearest tenth of a centimeter. Determine: the mean and standard deviation of the sampling distribution of ; the number of sample means that fall between 172.5 and 175.8 cm inclusive. (Hint: First compute the probability that a sample mean is between 172.5 and 175.8. Then multiply that number by the number of samples, 200).

Explanation / Answer

Mean ( u ) =174.5
Standard Deviation ( sd )=6.9/ Sqrt ( 25 )=1.38
Number ( n ) = 25
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  

To find P(a <= Z <=b) = F(b) - F(a)
P(X < 172.5) = (172.5-174.5)/6.9/ Sqrt ( 25 )
= -2/1.38
= -1.4493
= P ( Z <-1.4493) From Standard Normal Table
= 0.07363
P(X < 175.8) = (175.8-174.5)/6.9/ Sqrt ( 25 )
= 1.3/1.38 = 0.942
= P ( Z <0.942) From Standard Normal Table
= 0.82691
P(172.5 < X < 175.8) = 0.82691-0.07363 = 0.7533                  

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.