Using the Empirical Rule In Exercises 29–34, use the Empirical Rule.

The speeds for eight vehicles are listed. Using the sample statistics from Exercise 29, determine which of the data entries are unusual. Are any of the data entries very unusual? Explain your reasoning.
70, 78, 62, 71, 65, 76, 82, 64

검증된 단계별 안내

Step 1: Recall the Empirical Rule, which states that for a normal distribution: approximately 68% of the data falls within 1 standard deviation (σ) of the mean (μ), 95% within 2σ, and 99.7% within 3σ. Data points beyond 2σ are considered unusual, and those beyond 3σ are very unusual.

Step 2: Calculate the mean (μ) of the given data set. Use the formula: μ = (Σx) / n, where Σx is the sum of all data points and n is the number of data points.

Step 3: Calculate the standard deviation (σ) of the data set. Use the formula: σ = sqrt((Σ(x - μ)^2) / (n - 1)), where x represents each data point, μ is the mean, and n is the number of data points.

Step 4: Determine the range of usual data values using the Empirical Rule. Calculate μ ± 2σ for the range of usual values and μ ± 3σ for the range of very unusual values.

Step 5: Compare each data point (70, 78, 62, 71, 65, 76, 82, 64) to the calculated ranges. Identify which data points fall outside μ ± 2σ (unusual) and μ ± 3σ (very unusual). Provide reasoning based on these comparisons.

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주요 개념

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Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps in identifying how data is spread around the mean and is crucial for determining what constitutes 'usual' versus 'unusual' data points.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range. In the context of the Empirical Rule, standard deviation is used to calculate the ranges within which most data points fall.

Unusual Data Points

In statistics, data points are considered unusual if they lie beyond two standard deviations from the mean in either direction, which corresponds to the outer 5% of the data in a normal distribution. Identifying unusual data points is important for detecting outliers or anomalies that may require further investigation or could indicate errors in data collection.