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Feb 20, 2024

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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.
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