For Some Positive Value Of Z, The Probability That A Standar

For Some Positive Value Ofz The Probability That A Standard Normal Va

When dealing with standard normal distributions, knowledge of Z-scores and their associated probabilities is essential for statistical inference. The problem states that for some positive value of Z, the probability that a standard normal variable falls between 0 and Z is 0.3770. The task is to determine the value of Z corresponding to this probability.

In the standard normal distribution, the area to the right of zero is 0.5, and by symmetry, the area between zero and Z can be related to cumulative distribution function (CDF) values. The probability between 0 and Z of 0.3770 indicates that the CDF at Z minus 0.5 equals 0.3770 because the probability from 0 to Z is the difference between the cumulative probability at Z and the cumulative probability at 0 (which is 0.5). Therefore, P(0

Using standard normal distribution tables or software, we find the Z value that corresponds to a cumulative probability of 0.8770. Consulting the Z-table or a calculator, the value closest to 0.8770 is approximately 1.16. Hence, the Z-score corresponding to the probability that a standard normal variable falls between 0 and Z is 1.16.

Paper For Above instruction

The problem examines the probability distribution of a standard normal variable, focusing on the calculation of a specific Z-score based on a given probability. The standard normal distribution is a normal distribution with a mean of zero and a standard deviation of one, which is fundamental in many statistical analyses and hypothesis testing procedures.

Given that the probability that the variable Z lies between 0 and some positive value is 0.3770, we interpret this probability as the area under the standard normal curve between 0 and Z. To determine Z, we utilize the properties of the cumulative distribution function (CDF) for the standard normal, denoted as Φ(z). This function provides the probability that the variable is less than or equal to a particular Z-value.

Since the area from 0 to Z is 0.3770, the cumulative probability up to Z is obtained by adding this area to the probability from the extreme tail to 0, which is 0.5 (since the total probability from negative infinity to positive infinity is 1, and the distribution is symmetric around zero). Therefore, P(Z

This value indicates that Z = 1.16 is the point on the standard normal curve where the area from 0 to Z equals 0.3770, which matches the problem statement. This Z-score is critical in various applications, including confidence intervals, hypothesis testing, and probability calculations, serving as a benchmark for understanding the distribution's behavior.

References

  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
  • Moore, D. S., & McCabe, G. P. (2014). Introduction to the Practice of Statistics (8th ed.). W. H. Freeman and Company.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineers & Scientists (9th ed.). Pearson.
  • Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data (4th ed.). Pearson.
  • Larson, R., & Farid, A. (2017). Elementary Statistics. Pearson.
  • Neter, J., Wasserman, W., & Kutner, M. H. (1996). Applied Linear Statistical Models. McGraw-Hill.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Johnson, R. A., & Wichern, D. W. (2014). Applied Multivariate Statistical Analysis. Pearson.
  • Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). Norton.
  • Garrett, M. J. (2010). Understanding Probability. Cambridge University Press.

Related Assignments