In the example in problem 18, using a normal curve table, what is the minimum Z score an architect can have on the creativity test to be in the (a) top 50%, (b) top 40%, (c) top 60%, (d) top 30%, and (e) top 20%?
"
Oh, architects and their creativity! Let's calculate those Z scores with a touch of humor, shall we?
(a) To be in the top 50%, the minimum Z score for an architect is exactly zero! They just need to play it average, no pressure there.
(b) Moving up a notch, to be in the top 40%, the minimum Z score for our architect friend is around 0.25. Time to sprinkle a bit of creativity on that paper!
(c) For those aiming to be in the top 60%, the minimum Z score becomes around 0.84. We need a little more oomph, but our architects are up to the challenge.
(d) Pushing towards the top 30%, the minimum Z score jumps to approximately 0.52. Our architects need to put their best creative foot forward!
(e) Finally, for the ambitious souls aiming for the top 20%, the minimum Z score is about 0.84. Our architects must unleash their most imaginative and mind-blowing designs!
Remember, these Z scores are like little guides to help architects navigate the creativity spectrum. So, go forth and create masterpieces, my funny little builders of dreams!"
To determine the minimum Z score for different percentiles, we need to use a normal curve table. The table will give us the Z scores corresponding to the given percentiles. Here are the minimum Z scores for each percentile:
a) Top 50% - The median Z score is 0, which means 50% of the scores fall below it. Therefore, the minimum Z score to be in the top 50% is 0.
b) Top 40% - To find the Z score for the top 40%, we need to find the Z score that corresponds to a cumulative area of 0.60 (since the top 40% is the complement of the bottom 60%). Looking up in the table, the Z score corresponding to a cumulative area of 0.60 is approximately 0.253. So, the minimum Z score to be in the top 40% is 0.253.
c) Top 60% - This is the complement of the bottom 40%. The Z score that corresponds to a cumulative area of 0.40 is approximately -0.253 (since the table is symmetric about 0). So, the minimum Z score to be in the top 60% is -0.253.
d) Top 30% - To find the Z score for the top 30%, we need to find the Z score that corresponds to a cumulative area of 0.70 (since the top 30% is the complement of the bottom 70%). Looking up in the table, the Z score corresponding to a cumulative area of 0.70 is approximately 0.524. So, the minimum Z score to be in the top 30% is 0.524.
e) Top 20% - To find the Z score for the top 20%, we need to find the Z score that corresponds to a cumulative area of 0.80 (since the top 20% is the complement of the bottom 80%). Looking up in the table, the Z score corresponding to a cumulative area of 0.80 is approximately 0.841. So, the minimum Z score to be in the top 20% is 0.841.
Please note that these values are approximations as the table may differ slightly depending on the source.
To find the minimum Z score needed to be in a specific percentile using a normal curve table, you need to follow these steps:
Step 1: Look up the desired percentile in the body of the table. For example, if you want to find the minimum Z score for the top 50%, you would look up the value for 0.50.
Step 2: Once you find the desired percentile in the table, locate the corresponding Z score in the leftmost column of the table.
Now, let's find the minimum Z scores for various percentiles:
(a) Top 50%:
- Look up the value for 0.50 in the table.
- The closest value to 0.50 is 0.5000, which corresponds to a Z score of 0.
- Therefore, the minimum Z score to be in the top 50% is 0.
(b) Top 40%:
- Look up the value for 0.40 in the table.
- The closest value to 0.40 is 0.3997, which corresponds to a Z score of approximately -0.25.
- Therefore, the minimum Z score to be in the top 40% is approximately -0.25.
(c) Top 60%:
- Look up the value for 0.60 in the table.
- The closest value to 0.60 is 0.6010, which corresponds to a Z score of approximately 0.25.
- Therefore, the minimum Z score to be in the top 60% is approximately 0.25.
(d) Top 30%:
- Look up the value for 0.30 in the table.
- The closest value to 0.30 is 0.2995, which corresponds to a Z score of approximately -0.50.
- Therefore, the minimum Z score to be in the top 30% is approximately -0.50.
(e) Top 20%:
- Look up the value for 0.20 in the table.
- The closest value to 0.20 is 0.1985, which corresponds to a Z score of approximately -0.85.
- Therefore, the minimum Z score to be in the top 20% is approximately -0.85.
Remember, the normal curve table provides approximate values for the Z scores, so the values obtained may not be exact.