The Normal Distribution (The 68-95-99.7% Rule) PDF

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The Normal Distribution (The 68-95-99.7% Rule) The Normal Model for Bell -Shaped Distributions Numerical distributions that follow a bell -shape are said to be approximately normally distributed. Normal distributions are characterised by the mean 𝑥 and standard deviation 𝑠 of the distribution. The mean is placed at the centre (peak) of the bell curve with increasing standard deviations extending from either side. The bell curve is symmetric so percentages on either side will match and there is 50% above and below the mean. The 68-95-99.7% Rule The percentage of values that occur within standard deviations of the mean of approximately normally distributed distributions is approximately: • 68% of the data lies within one standard deviation of the mean • 95% of the data lies within two standard deviations of the mean • 99.7% of the data lies within three standard deviations of the mean These are extended to approximate the percentage of values around a particular standard deviation. The percentages can be added together to calculate the percentage greater than, less than, or between different standard deviations.

Example VCAA 2014 Exam 1 Question 2 The time spent by shoppers at a hardware store on a Saturday is approximately normally distributed with a mean of 31 minutes and a standard deviation of 6 minutes. If 2850 shoppers are expected to visit the store on a Saturday, the number of shoppers who are expected to spend between 25 and 37 minutes in the store is closest to 68% of 2850 = 1938.

Example VCAA 2001 Exam 1 Question 4 The distribution of systolic blood pressure of a large group of teenagers is approximately bell shaped with a mean of 122 and a standard deviation of 9. The percentage of these students with a systolic blood pressure less than 131 is approximately 0.15 + 2.35 + 13.5 + 34 + 34 = 84%. Example VCAA 2006 Exam 1 Question 4 The head circumference (in cm) of a population of infant boys is normally distributed with a mean of 49.5 cm and a standard deviation of 1.5 cm. Four hundred of these boys are selected at random and each boy’s head circumference is measured. The percentage of these boys with a head circumference of less than 48.0 cm is 0.15% + 2.35% + 13.5% = 16% The number of these boys with a head circumference of less than 48.0 cm is 16% of 400 = 64. Example VCAA 2019 Exam 1 Question 6 The time taken to travel between two regional cities is approximately normally distributed with a mean of 70 minutes and a standard deviation of 2 minutes. The percentage of travel times that are between 66 minutes and 72 minutes is closest to 13.5% + 34% + 34% = 81.5% Example VCAA 2018 Exam 1 Question 5 The pulse rates of a population of Year 12 students are approximately normally distributed with a mean of 69 beats per minute and a standard deviation of 4 beats per minute. The percentage of these students with a pulse rate of less than 61 beats per minute or greater than 73 beats per minute is closest to 0.15% + 2.35% + 13.5% + 2.35% + 0.15% = 18.5% A sample of 200 students was selected at random from this population. The number of these students with a pulse rate of less than 61 beats per minute or greater than 73 beats per minute is closest to 18.5% of 200 = 37...