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.

Use the same table as previous post.