A study of motor vehicle rates in the 50 states reveals
that traffic death rates (deaths per 100 million motor
vehicle miles driven) can be modeled by the normal
curve. The data suggest that the distribution has a mean
of 5.3 and a standard deviation of 1.3. Sketch the normal
curve, showing the mean and standard deviation.
The following link will give a good introduction to the concepts of a normal distributtion curve. Note that the article uses μ for mean, and σ for standard deviation.
http://en.wikipedia.org/wiki/Normal_distribution
To sketch the normal curve representing the traffic death rates, we need to understand the concept of a normal distribution and its characteristics.
The normal distribution, also known as the Gaussian distribution or the bell curve, is a continuous probability distribution that is symmetrical and has a characteristic bell shape. It is fully defined by its mean and standard deviation.
Mean (μ): The mean of a normal distribution represents the central value around which the data is distributed. In this case, the mean is given as 5.3 deaths per 100 million motor vehicle miles driven.
Standard Deviation (σ): The standard deviation of a normal distribution measures the spread or dispersion of the data. It indicates how far the data is likely to deviate from the mean. For this problem, the standard deviation is given as 1.3.
Now let's sketch the normal curve:
- Start by drawing a horizontal axis to represent the traffic death rates per 100 million motor vehicle miles driven.
- Choose an appropriate scale for the axis. Since the mean is 5.3 and the standard deviation is 1.3, you might consider labeling the axis from 0 (or lower if necessary) to 11 (or higher if necessary) to allow for potential deviations from the mean.
- Plot the mean (μ) on the horizontal axis. In this case, it would be located at 5.3.
- Mark one standard deviation (σ) above and below the mean. Since the standard deviation is 1.3, you would plot points at 5.3 + 1.3 = 6.6 and 5.3 - 1.3 = 4.0. These points represent one standard deviation above and below the mean.
- Connect the plotted points smoothly to form the bell-shaped curve. The highest point of the curve should be over the mean (5.3).
The resulting sketch should show a normal curve that is centered around the mean of 5.3 and has a spread of approximately 1.3 standard deviations in each direction.