Question

The heights of 1000 students are approximately normally distributed with a mean of 174.5centimeters and a standard deviation of 6.9 centimeters. Suppose 200 random samples ofsize 25 are drawn from this population and the means recorded to the nearest tenth of acentimeter. Determine

(a) the mean and standard deviation of the sampling distribution of ̄X;

(b) the number of sample means that fall between 171 and 177 cm .

Let X be a random variable following a continuous uniform distribution from 0 to 10. Findthe conditional probabilityP(X≥3|X <5.5).

2. Chebyshev’s theorem states that the probability that a random variableXhas a value atmost 3 standard deviations away from the mean is at least 8/9. Given that the probabilitydistribution of X is normally distributed with meanμand varianceσ2, find the exact value of

P(μ−3σ < X < μ+ 3σ)

Answer 1

Define random variable X : Heights of students.

X is normally distributed with mean = = 174.5 and standard deviation = = 6.9

Sample size = n = 25

a)

Here distribtion of random variable X is normal so sampling distribution of sample means is also normally distributed with mean = and standard deviation =

Mean of sampling distribution of :

Standard deviation of sampling distribution of :

Standard deviation of sampling distribution of :

b)

   where z is standard normal variable

  

= P(z < 1.81) - P(z < -2.54)

= 0.9649 - 0.0055 (From statistical table of z values)

= 0.9593

Number of sample means that fall between 171 and 177 cm = 200 * 0.9593 = 191.8619 = 192

Number of sample means that fall between 171 and 177 cm = 192