3. Describing Data Numerically
Standard Deviation
2:50 minutesProblem 3.2.33
Textbook Question
Identifying Significant Values with the Range Rule of Thumb. In Exercises 33–36, use the range rule of thumb to identify the limits separating values that are significantly low or significantly high.
U.S. Presidents Based on Data Set 22 “Presidents†in Appendix B, at the time of their first inauguration, presidents have a mean age of 55.2 years and a standard deviation of 6.9 years. Is the minimum required 35-year age for a president significantly low?
Verified step by step guidance1
Step 1: Recall the Range Rule of Thumb, which states that significantly low values are below μ - 2σ, and significantly high values are above μ + 2σ. Here, μ is the mean and σ is the standard deviation.
Step 2: Substitute the given values into the formulas for the significantly low and high limits. The mean (μ) is 55.2 years, and the standard deviation (σ) is 6.9 years. The formulas are:
Significantly Low Limit = μ - 2σ
Significantly High Limit = μ + 2σ.
Step 3: Calculate the significantly low limit using the formula:
Significantly Low Limit = 55.2 - 2(6.9).
Step 4: Calculate the significantly high limit using the formula:
Significantly High Limit = 55.2 + 2(6.9).
Step 5: Compare the minimum required age of 35 years to the significantly low limit. If 35 is below the significantly low limit, it is considered significantly low. Otherwise, it is not significantly low.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Range Rule of Thumb
The Range Rule of Thumb is a statistical guideline that suggests using the mean and standard deviation to identify significant values in a data set. According to this rule, values that fall more than two standard deviations away from the mean are considered significantly low or high. This helps in determining outliers and understanding the distribution of data.
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Mean and Standard Deviation
The mean is the average of a data set, calculated by summing all values and dividing by the number of values. The standard deviation measures the amount of variation or dispersion in a set of values. A low standard deviation indicates that values tend to be close to the mean, while a high standard deviation indicates a wider spread of values.
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Significance in Statistics
In statistics, a value is considered significant if it falls outside the expected range based on the mean and standard deviation. For example, in the context of the question, the minimum age of 35 years for a president can be evaluated against the calculated limits using the range rule of thumb to determine if it is significantly low compared to the average age of presidents.
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