Answer :
Final answer:
The 68-95-99.7 rule, also known as the empirical rule, describes how data behaves in a normal distribution. This principle is applicable in understanding large datasets such as fuel efficiencies of cars. A z-score of 'x' in a child's measurements would signify that the child is 'x' standard deviations from the mean of that particular measurement.
Explanation:
The 68-95-99.7 rule, also known as the empirical rule, describes how data behaves in a normal distribution. In a normal distribution:
- About 68% of the data falls within one standard deviation of the mean, between z-scores of -1 and +1.
- About 95% falls within two standard deviations of the mean, between z-scores of -2 and +2.
- About 99.7% falls within three standard deviations of the mean, between z-scores of -3 and +3.
For example, if we were looking at the fuel economy of a car model, and the mean fuel efficiency was 10, with a standard deviation of 3, we could use the z-scores to predict that about 95% of the cars will have a fuel efficiency between 4 and 16 (a z-score of ±2). Approximately 99.7% of cars will fall between a fuel efficiency of 1 and 19 (a z-score of ±3). This method gives us a very practical application of z-scores and the 68-95-99.7 rule in understanding large datasets such as fuel efficiencies amongst cars.
Regarding the child's 2-year checkup, if we learn that his z-score on some measurement is say, 2.0, this tells us that he is performing two standard deviations above the mean compared to other 2-year-olds on this same measurement. Similarly, a negative z-score would signify that the performance is below the mean.
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