Find the (two) z-scores that bound the middle 40% of the standard normal distribution. please show the work.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion and get your Z scores.
You are looking for .20 between the Z scores and the mean, one positive and the other negative.
To find the z-scores that bound the middle 40% of the standard normal distribution, we need to find the z-score for the lower bound and the z-score for the upper bound.
Step 1: Determine the area of the tail outside the middle 40%
The middle 40% represents the area between the lower bound and the upper bound. Since the standard normal distribution is symmetric, the area in each tail will be (100% - 40%) / 2 = 30%.
Step 2: Determine the z-score for the lower bound
To find the z-score for the lower bound, we need to find the cumulative probability that corresponds to 30%.
Using a standard normal distribution table, we can look up the z-score that corresponds to a cumulative probability of 0.30.
From the table, we find that the z-score corresponding to a cumulative probability of 0.30 is approximately -0.52.
So, the z-score for the lower bound is -0.52.
Step 3: Determine the z-score for the upper bound
To find the z-score for the upper bound, we need to find the cumulative probability that corresponds to 70% (30% + 40%).
From the standard normal distribution table, we find that the z-score corresponding to a cumulative probability of 0.70 is approximately 0.52.
So, the z-score for the upper bound is 0.52.
Therefore, the (two) z-scores that bound the middle 40% of the standard normal distribution are -0.52 and 0.52.
To find the z-scores that bound the middle 40% of the standard normal distribution, we need to determine the z-scores that correspond to the middle 20% and the middle 60% of the distribution.
Step 1: Find the z-score corresponding to the lower bound of the middle 20%.
Since the distribution is symmetrical, the lower bound of the middle 20% will be the 10th percentile. We can use a standard normal distribution table or a calculator to find this value.
Using a standard normal distribution table, we can find that the z-score corresponding to the 10th percentile is approximately -1.28.
Step 2: Find the z-score corresponding to the upper bound of the middle 20%.
The upper bound of the middle 20% will be the 90th percentile. Again, we can use a standard normal distribution table or a calculator to find this value.
Using a standard normal distribution table, we can find that the z-score corresponding to the 90th percentile is approximately 1.28.
Step 3: Find the z-score corresponding to the lower bound of the middle 60%.
The lower bound of the middle 60% will be the 20th percentile. We can again use a standard normal distribution table or a calculator to find this value.
Using a standard normal distribution table, we can find that the z-score corresponding to the 20th percentile is approximately -0.84.
Step 4: Find the z-score corresponding to the upper bound of the middle 60%.
The upper bound of the middle 60% will be the 80th percentile. We can use a standard normal distribution table or a calculator to find this value.
Using a standard normal distribution table, we can find that the z-score corresponding to the 80th percentile is approximately 0.84.
Therefore, the two z-scores that bound the middle 40% of the standard normal distribution are approximately -1.28 and 0.84.