Suppose that in the population of healthy children, VMA levels are Normally dist

Question

Suppose that in the population of healthy children, VMA levels are Normally distributed with mean 2.0 and standard deviation 0.3. Elevated VMA is defined as exceeding 2.3 mg. (Compute each probability below by hand and show your work.) i. What proportion of healthy children have elevated VMA levels? ii. Suppose that you take a random sample of 9 healthy children. What is the probability that their mean VMA is greater than 2.3 mg? iii. Suppose that you take a random sample of 25 healthy children. What is the probability that their mean VMA is greater than 2.3 mg? iv. The file VMA_densities.png shows the population distribution and each sampling distribution associated with questions (i)-(iii). Label each density with which question (i-iii) it is associated with, as well as the mean and SD of the Normal distribution it represents. For the probability in (i), draw the area-under-the-curve associated with this probability, using the density it is associated with.

Explanation / Answer

Result: Suppose that in the population of healthy children, VMA levels are Normally distributed with mean 2.0 and standard deviation 0.3. Elevated VMA is defined as exceeding 2.3 mg. (Compute each probability below by hand and show your work.)

z value = z =( 2.3-2.0)/0.3 =1

P( x > 2.3) = p( z > 1) =0.1587

Standard error = sd/sqrt(n) = 0.3/sqrt(9) =0.1

P( mean x > 2.3) = p( z > 0.1) = 0.4602

Standard error = sd/sqrt(n) = 0.3/sqrt(25) =0.06

P( mean x > 2.3) = p( z > 0.06) = 0.4761

iv. The file VMA_densities.png shows the population distribution and each sampling distribution associated with questions (i)-(iii). Label each density with which question (i-iii) it is associated with, as well as the mean and SD of the Normal distribution it represents. For the probability in (i), draw the area-under-the-curve associated with this probability, using the density it is associated with.

1). Mean = 2.0, sd= 0.3

2). Mean = 2.0, sd= 0.1

3). Mean = 2.0, sd= 0.06

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