3. Describing Data Numerically
Interpreting Standard Deviation
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A sample of 500 random adult books in a library has an average of 312 pages with a standard deviation of 26 pages. Find the percentage of books in the sample with less than 338 pages using the Empirical Rule of Standard Deviation.
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Verified step by step guidance1
Identify the key information provided in the problem: the sample mean (312 pages), the standard deviation (26 pages), and the value of interest (338 pages). The problem also specifies the use of the Empirical Rule.
Recall the Empirical Rule: Approximately 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This rule applies to data that follows a normal distribution.
Calculate how many standard deviations the value of 338 pages is from the mean. Use the formula: z = (X - μ) / σ, where X is the value of interest (338), μ is the mean (312), and σ is the standard deviation (26).
Using the z-score calculated in the previous step, determine the percentage of data below this z-score. Since the Empirical Rule provides approximate percentages, interpret the z-score in terms of the rule (e.g., if the z-score is 1, approximately 84% of the data falls below this value).
Finally, match the percentage obtained from the Empirical Rule to the closest answer choice provided in the problem.
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