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Subject
Mathematics
Date
Feb 20, 2024
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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%
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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.
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