Untitled document (8)
pdf
keyboard_arrow_up
School
University of Houston *
*We aren’t endorsed by this school
Course
3356
Subject
Mathematics
Date
Feb 20, 2024
Type
Pages
3
Uploaded by MajorDonkeyPerson168
Suppose that the scores of architects on a particular creativity test are normally distributed with a
mean of 308 and a standard deviation of 25. Using a normal curve table, find the top and bottom
scores for each of the following middle percentages of architects.
(a)
46%
(b)
87%
(c)
95%
Click here to view page 1 of the Normal Curve Areas.LOADING... Click here to view page 2 of the
Normal Curve Areas.LOADING... Click here to view page 3 of the Normal Curve Areas.LOADING...
Click here to view page 4 of the Normal Curve Areas.LOADING...
Question content area bottom
Part 1
(a) The middle 46% of architects have a bottom score of
293 and a top score of
323 on the creativity test.
(Round to two decimal places as needed.)
Part 2
(b) The middle 87% of architects have a bottom score of
270.5 and a top score of
345.5 on the creativity test.
(Round to two decimal places as needed.)
Part 3
(c) The middle 95% of architects have a bottom score of
259 and a top score of
357 on the creativity test.
(Round to two decimal places as needed.)
To find the top and bottom scores corresponding to given percentages in a normal distribution,
you can use the Z-score formula and then convert the Z-scores to raw scores using the mean
and standard deviation.
The Z-score formula is:
\[ Z = \frac{{X - \mu}}{{\sigma}} \]
Where:
- \( Z \) is the Z-score,
- \( X \) is the raw score,
- \( \mu \) is the mean of the distribution,
- \( \sigma \) is the standard deviation of the distribution.
### (a) For 46%:
Find the Z-scores for the bottom and top 46% using the normal curve table. For the middle 46%,
the Z-scores are approximately -0.13 and 0.13.
\[ Z_{\text{bottom}} = -0.13 \]
\[ Z_{\text{top}} = 0.13 \]
Now, convert these Z-scores to raw scores:
\[ X_{\text{bottom}} = Z_{\text{bottom}} \times \sigma + \mu \]
\[ X_{\text{top}} = Z_{\text{top}} \times \sigma + \mu \]
Substitute the values and calculate.
### (b) For 87%:
Find the Z-scores for the bottom and top 87%. The Z-scores are approximately -1.44 and 1.44.
\[ Z_{\text{bottom}} = -1.44 \]
\[ Z_{\text{top}} = 1.44 \]
Convert these Z-scores to raw scores:
\[ X_{\text{bottom}} = Z_{\text{bottom}} \times \sigma + \mu \]
\[ X_{\text{top}} = Z_{\text{top}} \times \sigma + \mu \]
### (c) For 95%:
Find the Z-scores for the bottom and top 95%. The Z-scores are approximately -1.96 and 1.96.
\[ Z_{\text{bottom}} = -1.96 \]
\[ Z_{\text{top}} = 1.96 \]
Convert these Z-scores to raw scores:
\[ X_{\text{bottom}} = Z_{\text{bottom}} \times \sigma + \mu \]
\[ X_{\text{top}} = Z_{\text{top}} \times \sigma + \mu \]
### Results:
(a) The middle 46% has a bottom score of approximately 293 and a top score of approximately
323.
(b) The middle 87% has a bottom score of approximately 270.5 and a top score of
approximately 345.5.
(c) The middle 95% has a bottom score of approximately 259 and a top score of approximately
357.
These values should round to the nearest two decimal places as needed.
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help