Problem 16 Points: Calculate The Area Under The Normal Curve

Problem 16 Points Calculate The Area Under The Normal Curve For The

Calculate the area under the normal curve for the following values of X, when the mean is 100 and the standard deviation is 20.

(a.) P(X > 140)

(b.) P(130

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The calculation of areas under the normal distribution curve is fundamental in statistics, particularly when assessing probabilities associated with normally distributed variables. Given a mean (μ) of 100 and a standard deviation (σ) of 20, we evaluate the probabilities for specified intervals using standard Z-scores and corresponding standard normal distribution tables or computational tools.

Part (a): P(X > 140)

To find this probability, first compute the Z-score for X=140:

Z = (X - μ) / σ = (140 - 100) / 20 = 40 / 20 = 2

The probability P(X > 140) equals the area to the right of Z=2 under the standard normal curve. Consulting standard normal distribution tables or using software, the cumulative probability up to Z=2 is approximately 0.9772. Therefore:

P(X > 140) = 1 - P(Z ≤ 2) = 1 - 0.9772 = 0.0228

Part (b): P(130

Calculate Z-scores for both bounds:

Z1 = (130 - 100) / 20 = 30 / 20 = 1.5


Z2 = (140 - 100) / 20 = 2 (as calculated earlier)

Using the standard normal table or software:

P(Z 

P(Z

The probability that X falls between 130 and 140 is:

P(130

These calculations demonstrate how Z-scores and the standard normal distribution facilitate the determination of probabilities for specific ranges of normally distributed variables, enabling statistical inference and decision-making processes.

References

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