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.
We cannot provide a sketch here.
To sketch the normal curve and depict the mean and standard deviation, you can follow these steps:
Step 1: Understand the parameters
The problem provides two pieces of information about the distribution:
- Mean (μ) = 5.3
- Standard deviation (σ) = 1.3
Step 2: Plot the x-axis
The x-axis represents the range of values for the random variable. In this case, the random variable is the traffic death rate. Choose a suitable range for the x-axis that covers a sufficient span of values based on the context. Let's assume a range from 0 to 10 for this problem.
Step 3: Determine the z-scores for σ
The z-score is a measure of how many standard deviations a given value lies from the mean. The formula to calculate the z-score is:
z = (x - μ) / σ
In this case, we use the standard deviation (σ) to calculate the z-score. Since we are interested in one standard deviation above and below the mean, we need to calculate the z-scores for ± 1.
z1 = (5.3 - 1.3) / 1.3 = 3
z2 = (5.3 + 1.3) / 1.3 = 7
Step 4: Plot the normal curve
The normal curve is a symmetric bell-shaped curve centered around the mean. The proportion of values falling within one standard deviation of the mean is approximately 68%.
On the x-axis, plot the z-scores obtained in step 3. So, 3 would be towards the left side and 7 towards the right side. Place the mean (μ) at 5.3.
Step 5: Mark standard deviation values
Draw vertical lines from each z-score (3 and 7) to intersect with the curve. These lines represent one standard deviation above and below the mean. Label them as "+1σ" and "-1σ," respectively.
Step 6: Illustrate the normal curve
Sketch a smooth curve that starts slightly below the mean, peaks at the mean (μ), and then descends symmetrically to slightly above the mean on the other side. The curve should look like a symmetric bell shape.
Remember that the actual shape may vary slightly depending on the proportion of data within different standard deviations from the mean.
By following these steps, you should be able to sketch the normal curve with the mean and standard deviation for the given data.