What is the highest z-value for the lowest 10% of observations on the standard normal distribution?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.10) and relate to the Z score. Remember that, since it is below the mean, it will be negative.
-1.2816
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To find the highest z-value for the lowest 10% of observations on the standard normal distribution, we need to use the concept of the cumulative distribution function (CDF). The CDF gives the probability that a random variable takes a value less than or equal to a given value.
Here's how you can find the answer:
1. Determine the area under the standard normal curve that corresponds to the lowest 10% of observations. Since we want the highest z-value for this area, we need to find the z-value that corresponds to the upper tail (1 - 0.10 = 0.90).
2. Look up the z-value in a standard normal distribution table or use a calculator that provides the cumulative probability (CDF) function for the standard normal distribution.
Using a standard normal distribution table:
- Locate the value closest to 0.90 in the table.
- Find the corresponding z-value in the intersecting row and column. This z-value represents the highest z-value for the lowest 10% of observations.
Using a calculator:
- Utilize the inverse CDF function. For example, in Python, you can use the scipy.stats.norm.ppf() function with a probability argument of 0.90 to calculate the z-value.
The resulting z-value represents the highest z-score for the lowest 10% of observations on the standard normal distribution.