Given Z ~ N(0, 1) Calculate: 1.1 P (-2.01 < Z < 2.5) 1.2 P

Given Z ~ N (0, 1) Calculate: 1.1 P (- 2.01 1..3 1.4 1.5 1.6 1.7 1.8 1.9

Calculate the probabilities associated with a standard normal distribution Z, which has a mean of 0 and a standard deviation of 1.

Paper For Above instruction

The standard normal distribution, denoted as Z ~ N(0,1), is a foundational concept in inferential statistics. It allows researchers and statisticians to understand the probabilities or proportions of data points falling within specific ranges, which is essential for hypothesis testing, confidence intervals, and other statistical inferences. This paper explores the calculation of specific probabilities related to the standard normal distribution, particularly focusing on the probability that a standard normal variable falls within certain bounds.

Understanding the Standard Normal Distribution

The standard normal distribution is symmetrical around its mean of zero, with its shape determined by the standard deviation of one. The probability density function (PDF) describes the likelihood of a random variable falling within a particular range. To find probabilities associated with the standard normal distribution, statisticians typically use the cumulative distribution function (CDF), which gives the probability of the variable being less than or equal to a specific value.

Calculations of Probabilities

The first probability, P(-2.01 , involves calculating the probability that Z falls between -2.01 and 2.5. This can be expressed using the standard normal CDF as:

P(-2.01

where Φ(x) denotes the cumulative probability up to x.

Using standard normal distribution tables or statistical software such as R, Python, or dedicated calculator functions, we can obtain these values:

  • Φ(2.5) ≈ 0.9938
  • Φ(-2.01) ≈ 0.0222

Thus, the probability that Z is between -2.01 and 2.5 is approximately:

0.9938 - 0.0222 = 0.9716

The second probability, P(Z > 1.3), measures the likelihood that Z is greater than 1.3. This can be directly calculated as:

P(Z > 1.3) = 1 - Φ(1.3)

From the standard normal table or software:

  • Φ(1.3) ≈ 0.9032

Therefore:

1 - 0.9032 = 0.0968

Similarly, for P(Z , the calculation involves the cumulative probability up to -2.4:

P(Z

For the probability that Z is less than -2.5, the calculation is:

P(Z

Next, to find the probability that Z lies between 0.48 and 2.6, the calculation is:

P(0.48

  • Φ(2.6) ≈ 0.9953
  • Φ(0.48) ≈ 0.6844

Thus:

0.9953 - 0.6844 = 0.3109

Finally, the probability that Z is less than -2.85 is:

Φ(-2.85) ≈ 0.0022

And the probability that Z exceeds -2.75 is:

1 - Φ(-2.75) ≈ 1 - 0.0029 = 0.9971

These calculations exemplify how standard normal probabilities are derived using tables or software, enabling precise statistical inference in various research scenarios. Such computations are fundamental in quality control, finance, psychology, and many other fields that rely on normal distribution assumptions.

References

  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Brooks/Cole, Cengage Learning.
  • Mendenhall, W., Beaver, R. J., & Beaver, B. M. (2012). Introduction to Probability and Statistics (14th ed.). Brooks/Cole.
  • Blitzstein, J., & Hwang, J. (2014). Introduction to Probability. CRC Press.
  • Schervish, M. J. (2012). Theory of Statistics. Springer.
  • Wilks, S. S. (2011). Mathematical Statistics. CRC Press.
  • Rice, J. A. (2006). Mathematical Statistics and Data Analysis. Cengage Learning.
  • NIST/SEMATECH. (2012). e-Handbook of Statistical Methods. National Institute of Standards and Technology.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
  • Hogg, R. V., & Tanis, E. (2010). Probability and Statistical Inference (9th ed.). Pearson.
  • Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data. Pearson.

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