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(b) What percentage of standardized test scores is less than 183 or greater than \( 867 ? \) \( \% \) (Round to one decimal place as needed.)

Ask by Sandoval Lawson. in the United States
Feb 22,2025

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Answer

31.7% of standardized test scores are either less than 183 or greater than 867.

Solution

We are given that 183 and 867 are exactly one standard deviation below and above the mean, respectively. That is, the mean is   (183 + 867) / 2 = 525, and one standard deviation is   525 – 183 = 342  or  867 – 525 = 342. For a normal distribution, the probability that a score is less than one standard deviation below the mean (z = –1) is about 15.87%, and the probability of a score being greater than one standard deviation above the mean (z = +1) is also about 15.87%. Thus, the total percentage for scores that are either less than 183 or greater than 867 is   15.87% + 15.87% = 31.74%. Rounded to one decimal place, the percentage is 31.7%.

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To find the percentage of standardized test scores that is less than 183 or greater than 867, we need to know the distribution of the scores. Generally, if the scores follow a normal distribution, we can calculate the z-scores for 183 and 867 and then use the standard normal distribution table to find the corresponding percentages. Assuming a mean (μ) and standard deviation (σ) for the scores, calculate the z-scores: \( z = \frac{{x - μ}}{σ} \) Once you determine the z-scores, look them up in the standard normal distribution table to find the probabilities. The percentage you seek is then the sum of the probabilities for the two tails of the distribution corresponding to those z-scores. Don't forget to round your final answer to one decimal place.

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