Find the value of Z that is smaller than 90% of all values of Z.
To find the value of Z that is smaller than 90% of all values of Z, you will need to use percentiles. A percentile represents a specific value below which a given percentage of the data falls.
Step 1: Determine the desired percentile. In this case, we want to find a value that is smaller than 90% of all values, so the desired percentile is 90.
Step 2: Convert the desired percentile to a decimal. Divide the desired percentile by 100:
90 ÷ 100 = 0.9
Step 3: Subtract the desired percentile (expressed as a decimal) from 1. This gives you the complement to the desired percentile:
1 - 0.9 = 0.1
Step 4: Use statistical tools or software to find the Z-score associated with the complement of the desired percentile. The Z-score represents the number of standard deviations a value is away from the mean in a normal distribution.
For a standard normal distribution, the Z-score associated with the complement of 0.1 is approximately -1.28.
Step 5: Use the Z-score formula to find the corresponding value of Z:
Z = (Z-score * standard deviation) + mean
Note: The mean of a standard normal distribution is 0, and the standard deviation is 1.
Z = (-1.28 * 1) + 0
Z = -1.28
Therefore, the value of Z that is smaller than 90% of all values of Z is approximately -1.28.
To find the value of Z that is smaller than 90% of all values of Z, you would need a dataset or a distribution of values for Z. Without specific data or a distribution, it is not possible to determine the exact value. However, I can explain the general process of finding a value in a dataset that is smaller than a certain percentage of the values:
1. Arrange the dataset or values for Z in ascending order.
2. Calculate the percentile corresponding to the desired percentage. In this case, the desired percentage is 90%, so the percentile would be 90.
3. Determine the position in the sorted dataset that corresponds to the desired percentile. This can be calculated using the formula: position = (percentile / 100) * (number of observations + 1).
4. If the position is an integer, the value at that position corresponds to the desired value of Z. If the position is a decimal, you may need to interpolate between the neighboring values to estimate the value of Z.
5. Alternatively, you can use statistical software or a calculator that provides functions for calculating percentiles to find the exact value.
Remember that without specific data or a distribution, it is not possible to provide an exact value for Z that is smaller than 90% of all values of Z.