Find The Value Of Z For The Shaded Region 364269

Find the value of z for which the area of the shaded region under the standard normal curve is as specified

Understand the problem: The task involves finding the z-score associated with a specific area under the standard normal distribution curve. The standard normal distribution is a symmetric probability distribution with a mean of 0 and a standard deviation of 1. The area to the left of a particular z-score corresponds to the probability of a random variable being less than that z-score.

Given data: The areas specified are 0.0026, 0.0013, 0.383, and 0.6915, each representing the cumulative area from the left under the standard normal curve up to the z-value that we need to find. Our goal is to determine the z-scores corresponding to these areas.

Method: To find z for a given area, we use the inverse cumulative distribution function (inverse CDF), also known as the z-score table or the percentile function (invNorm in many calculators or statistical software). The inverse CDF gives us the z-value for a given cumulative probability.

Calculations:

1. For area = 0.0026:

Using a z-table or a calculator, find z such that P(Z invNorm(0.0026) ≈ -2.81 or similar, depending on the table or software precision.

2. For area = 0.0013:

Similarly, find z such that P(Z -3.00.

3. For area = 0.383:

This area is just less than 0.5, corresponding to a negative z-value. Approximately, -0.30.

4. For area = 0.6915:

This area is greater than 0.5, corresponding to a positive z-value. Approximate z: 0.50.

Final answers:

  • z ≈ -2.81 for area 0.0026
  • z ≈ -3.00 for area 0.0013
  • z ≈ -0.30 for area 0.383
  • z ≈ 0.50 for area 0.6915

Summary:

The calculation of z-values corresponding to specified areas under the standard normal curve is accomplished using the inverse cumulative distribution function. These z-scores are useful in standardizing and analyzing probabilistic data in statistics.

References

  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
  • Moore, D. S., & McCabe, G. P. (2009). Introduction to the Practice of Statistics (6th ed.). W.H. Freeman.
  • Wolfram Research. (2020). Wolfram Alpha Math World: Standard Normal Distribution. Wolfram Research.
  • NIST/SEMATECH. (2012). e-Handbook of Statistical Methods. National Institute of Standards and Technology.
  • Zwillinger, D. (2003). CRC Standard Mathematical Tables and Formulae (32nd ed.). CRC Press.

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