In each case, sketch the two specified normal curves on the same set of axes:
a A normal curve with m 20 and s 3, and a normal curve with m 20 and s 6.
b A normal curve with m 20 and s 3, and a normal curve with m 30 and s 3.
c A normal curve with m 100 and s 10, and a normal curve with m 200 and s 20.
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To sketch the two specified normal curves on the same set of axes, we need to understand the key characteristics of a normal distribution: the mean (μ) and standard deviation (σ). The mean represents the center of the distribution, and the standard deviation determines the spread or width of the curve.
a) For the first case, we are given two normal curves: one with mean (m) of 20 and standard deviation (s) of 3, and the other with a mean of 20 and standard deviation of 6.
To start sketching, we plot the first normal curve. The mean of 20 is the center point, so we mark it on the x-axis. Since the standard deviation is 3, we go 3 units to the right and left of the mean and mark these points on the x-axis. These marks represent the points where the curve starts to descend.
Next, we consider the second normal curve. We use the same mean of 20 as a reference point on the x-axis. However, this time the standard deviation is 6. So, we go 6 units to the right and left of the mean (20) and mark these points on the x-axis as the starting points of the curve.
The resulting sketch should show two normal curves centered at 20, but with different spreads. The first curve with a standard deviation of 3 will be narrower than the second curve, which has a standard deviation of 6.
b) In this case, we have a normal curve with a mean of 20 and a standard deviation of 3, and another normal curve with a mean of 30 and a standard deviation of 3.
We start by plotting the first normal curve. The mean of 20 serves as the center point on the x-axis. The standard deviation is 3, so we go 3 units to the right and left of the mean and mark these points on the x-axis.
Next, we consider the second normal curve. It has a mean of 30, which we mark on the x-axis. Similarly, we go 3 units to the right and left of this mean (30) and mark these points on the x-axis.
The resulting sketch should show two normal curves, one centered at 20 and the other at 30, with the same spread (standard deviation of 3).
c) For the last case, we have a normal curve with a mean of 100 and a standard deviation of 10, and another normal curve with a mean of 200 and a standard deviation of 20.
We begin by plotting the first normal curve. The mean of 100 is the center point on the x-axis. The standard deviation is 10, so we go 10 units to the right and left of the mean and mark these points on the x-axis.
Next, we consider the second normal curve. Its mean is 200, which we mark on the x-axis. We go 20 units to the right and left of this mean (200) and mark these points on the x-axis.
The resulting sketch should show two normal curves, one centered at 100 and the other at 200, with the second curve being wider than the first due to its larger standard deviation (20 compared to 10).