MA 121 (mean and standard deviation)


#1

I am having difficulty arriving at this answer. It makes sense, although I am not sure how to break this out by step…Please Help

Question: A continuous random variable X has a normal distribution with mean 100 and standard deviation 10. Sketch a qualitatively accurate graph of its density function.

Answer: The graph is a bell-shaped curve centered at 100 and extending from about 70 to 130


#2

I believe I figured it out. You simply multiply each deviation in increments, (with the exception of the mean alone) beginning with mean + sigma, mean + sigma(2), mean + sigma(3), etc. The reverse holds true for the negative side of the mean (mean - sigma, mean - sigma(2), etc.)


#3

Properties of normal curve.
Normal curve has bell shaped and the curve is symmetric about it’s mean. The curve is completely determined by the mean (μ) and the standard deviation (σ). The area under the curve to the right of the mean is 0.5 and the area under the curve to the left of the mean is 0.5.
According to empirical rule
• approximately 68% of the measurements will fall within 1 standard deviations of the mean, i.e. within the interval ( μ− 1σ, μ + 1σ)
• approximately 95% of the measurements will fall within 2 standard deviations of the mean, i.e. within the interval ( μ− 2σ, μ + 2σ)
• approximately 99.7% of the measurements will fall within 3 standard deviations of the mean, i.e. within the interval ( μ− 3σ, μ + 3σ)
Using all those properties the normal curve for X variable with mean( μ ) 100 and standard deviation (σ) 10 is
image