(GRAPH) Sketch a normal curve that has a mean of 15 and a standard deviation of 4. On the same x-axis, sketch another normal curve that has a mean of 25 and a standard deviation of 4. Describe the two normal curves.
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That means the center of the x-axis on your normal curve has the number 15 (the mean of 15), then 1 z score to the left is the number 11, and one to the right of the 15 is 19. We know that 68% of all the data pieces lay within one standard deviation of the mean. Continue outward for three standard deviations. Along the x-axis begin a curve that has a mean of 25 (so you will see that the two normal curves are side by side on the cartesian plane).
the graph pleaseeee.
i want to know the sketch normal curve that has a mean of 15 and a standard deviation of 4
Well, let's get graphically funny, shall we? Picture a stage set with two performers.
The first performer's name is "Mean 15". He's got a wonderfully average height of 15, and he's not too rigid or too flexible, represented by a standard deviation of 4. He is the epitome of the classic normal curve, symmetrical and bell-shaped. His performances are reliable and predictable, with most of them happening right around his mean, 15. The crowd loves his consistency!
Now, let's move on to the second performer, "Mean 25". This guy is a bit taller, standing at a mean of 25. He shares the same standard deviation of 4, so he's equally flexible as the first performer. However, since he stands taller, his performances are slightly shifted to the right on the x-axis. His curve is also symmetric and bell-shaped, just like the first performer’s, but higher on the stage.
So, in summary, we have two normal curves representing performers with different averages. "Mean 15" is standing at 15, while "Mean 25" stands at 25. Both performers share the same level of flexibility (standard deviation of 4), but due to their different means, their performances are shifted on the x-axis. It's like watching two acrobats, one expertly hitting the standard mark, and the other going for something higher!
To sketch the normal curves, we can follow these steps:
1. Determine the x-axis range: Since the standard deviation is 4, we can start by selecting an x-axis range that covers multiple standard deviations on both sides of the mean. Let's select a range of -3σ to +3σ, which in this case would be (-3 * 4) + 15 = 3 to (3 * 4) + 15 = 27.
2. Calculate the y-axis values for each x-value: We need to calculate the y-axis values for each x-value using the formula for the normal distribution, which is given by:
y = (1 / (√(2π) * σ)) * e^(-((x - μ)^2) / (2σ^2))
Where:
- μ is the mean,
- σ is the standard deviation,
- π is a mathematical constant (approximately 3.14159),
- e is the mathematical constant base of natural logarithms (approximately 2.71828).
For the first normal curve with mean μ₁ = 15 and standard deviation σ₁ = 4, the formula becomes:
y₁ = (1 / (4 * √(2π))) * e^(-((x - 15)^2) / 32)
Similarly, for the second normal curve with mean μ₂ = 25 and standard deviation σ₂ = 4, the formula becomes:
y₂ = (1 / (4 * √(2π))) * e^(-((x - 25)^2) / 32)
3. Plot the curves: Using the calculated y-values, we can plot the points on the graph and connect them to visualize the normal curves.
The first normal curve (mean of 15 and standard deviation of 4) will be centered around x = 15, and the second normal curve (mean of 25 and standard deviation of 4) will be centered around x = 25. Both curves will have the same spread, determined by the standard deviation.
The first curve will have a peak at x = 15 and a relatively lower spread compared to the second curve. It will gradually decrease on each side of the mean as the x-values move further away from 15.
The second curve will have a peak at x = 25 and the same spread as the first curve. It will also decrease on each side of the mean as the x-values move further away from 25.
By sketching these curves on the same x-axis, you can visually compare their shapes and see how they differ based on their means and standard deviations.