Question

Let X e N(50,36). Using the same set of axes, sketch thegraphs of the probability density functions of

a) X
b) X, the mean of a random sample of size 9 fromthis distribution
c) X, the mean of a random sample of size 36 from thisdistribution

Answer 1

To sketch the probability density functions, plot the graph of the probability density function of X, the mean of a random sample of size 9, and the mean of a random sample of size 36 from a normal distribution N(50, 36). Use the central limit theorem to adjust the standard deviation for the mean of the random samples.

Step-by-step explanation:

To sketch the probability density functions of X, the mean of a random sample of size 9, and the mean of a random sample of size 36 from a normal distribution N(50, 36), follow these steps:

1. Plot the graph of the probability density function of X, which will be a bell-shaped curve centered around the mean of 50.
2. To sketch the graph of the probability density function of X, the mean of a random sample of size 9, use the central limit theorem. The mean of X will also be 50, but the standard deviation will be smaller, equal to the standard deviation of the population divided by the square root of the sample size, so 36/√9 = 12.
3. Finally, to sketch the graph of the probability density function of X, the mean of a random sample of size 36, the standard deviation will be even smaller, equal to 36/√36 = 6.