A random sample is obtained from a normal population with a mean of μ = 95 and a standard deviation of σ = 40. The sample mean is μ = 86.1) Is this a representative sample mean or an extreme value for a sample of n = 16 scores? 2) Is this a representative sample mean or an extreme value for a sample of n = 100 scores?

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Mcdowell

Mathematics

1 November, 04:29

A random sample is obtained from a normal population with a mean of μ = 95 and a standard deviation of σ = 40. The sample mean is μ = 86.1) Is this a representative sample mean or an extreme value for a sample of n = 16 scores? 2) Is this a representative sample mean or an extreme value for a sample of n = 100 scores?

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Pumpkin · Answer 1

-0.89 is within the central 95% of the distribution, it is neither extreme or unusual.

-2.225 is outside the central 95% of the distribution, it is extreme and unusual

Step-by-step explanation:

We have to:

μ = 95

σ = 40

M = 86.1

The central 95% of the unit's normal distribution is between z = ± 1.96. Therefore if it is within this range it is not extreme, nor unusual; but if it comes out of this range it is extreme and unusual.

Case 1.

n = 16

The formula to use is the following:

z-score = (M - μ) / σS

Where σS = σ / (n ^ (1/2))

Replacing the values:

σS = 40 / (16 ^ (1/2)) = 10

z-score = (86.1 - 95) / 10 = - 0.89

Since this z-score - 0.89 is within the central 95% of the distribution (± 1.96), it is neither extreme or unusual.

Case 2.

n = 100

Replacing the values:

σS = 40 / (100 ^ (1/2)) = 4

z-score = (86.1 - 95) / 4 = - 2.225

Since this z-score - 2.225 is outside the central 95% of the distribution (± 1.96), it is extreme and unusual.