The z-scores that separate the middle 52% of the distribution from the area in the tail of the standard normal distribution are approximately Z1 ≈ -0.7112 and Z2 ≈ 0.6745.
To find the z-scores that separate the middle 52% of the distribution from the area in the tail of the standard normal distribution, you can follow these steps:
1. Understand the problem:
You want to find two z-scores, let's call them Z1 and Z2, such that the middle 52% of the standard normal distribution is between these two z-scores, and the remaining area (tail) is outside them.
2. Find the cumulative probabilities:
To do this, you need to find the cumulative probabilities associated with the middle 52% and the tail areas.
a) Middle 52%:
To find the middle 52%, you can start by finding the cumulative probability for the lower percentile. Since the middle 52% is symmetric around the mean (z = 0), you need to find the cumulative probability for the lower 24% (i.e., (100% - 52%) / 2 = 24%).
You can use a standard normal distribution table or a calculator to find this value. In many cases, you'll find the value for the 24th percentile to be approximately 0.2438.
b) Tail area:
The remaining area in the tails is 100% - 52% = 48%. Since it's evenly split between the two tails, each tail has an area of 24%.
3. Find the z-scores:
Now that you have the cumulative probability for the lower 24% (0.2438) and the tail area of 24%, you can use the z-table or a calculator to find the z-scores associated with these probabilities.
a) For the lower 24%, you can look up the z-score corresponding to the cumulative probability of 0.2438. This will give you Z1.
b) For the upper 24% (tail area), you can look up the z-score corresponding to the cumulative probability of 1 - 0.24 = 0.76. This will give you Z2.
4. Calculate Z1 and Z2:
Using a standard normal distribution table or calculator, you can find the z-scores for the probabilities you found:
a) Z1 ≈ -0.7112 (for the lower 24%)
b) Z2 ≈ 0.6745 (for the upper 24%)