Question

name lets be normal learning task until this task, we have focused on distributions of discrete data. we will now direct our attention to continuous data. where a discrete variable has a finite number of possible values, a continuous variable can assume all values in a given interval of values. therefore, a continuous random variable can assume an infinite number of values. we will focus our attention specifically on continuous random variables with distributions that are approximately normal. remember that normal distributions are symmetric, bell - shaped curves that follow the empirical rule. the empirical rule for normal distributions states that • 68% of the data values will fall within one standard deviation of the mean, • 95% of the data values will fall within two standard deviations of the mean, and • 99.7% of the data values will fall within three standard deviations of the mean. in this activity, we will use probability histograms and approximate these histograms to a smooth curve that displays the shape of the distribution without the boxiness of the histogram. we will also assume that all of the data we use is approximately normally distributed. review: 1) the distribution of heights of adult american women is approximately normal with a mean of 65.5 inches (μ = 65.5) and a standard deviation of 2.5 inches (σ = 2.5). draw a normal curve and label the mean and points one, two, and three standard deviations above and below the mean. this work is licensed under a creative commons attribution - noncommercial - sharealike 4.0 international license © 2023 georgia department of education, all rights reserved last updated july 2024 page 2 of 17

Explanation:

Step1: Calculate values for one - standard deviation

One standard deviation above the mean: $\mu+\sigma = 65.5 + 2.5=68$ inches.
One standard deviation below the mean: $\mu - \sigma=65.5 - 2.5 = 63$ inches.

Step2: Calculate values for two - standard deviations

Two standard deviations above the mean: $\mu + 2\sigma=65.5+2\times2.5 = 65.5 + 5=70.5$ inches.
Two standard deviations below the mean: $\mu - 2\sigma=65.5-2\times2.5=65.5 - 5 = 60.5$ inches.

Step3: Calculate values for three - standard deviations

Three standard deviations above the mean: $\mu+3\sigma=65.5 + 3\times2.5=65.5+7.5 = 73$ inches.
Three standard deviations below the mean: $\mu - 3\sigma=65.5-3\times2.5=65.5 - 7.5 = 58$ inches.

Step4: Draw the normal curve

Draw a bell - shaped curve. Mark the mean $\mu = 65.5$ at the center of the curve. Mark the points 58, 60.5, 63, 65.5, 68, 70.5, 73 on the horizontal axis corresponding to three, two, one standard deviations below the mean, the mean, and one, two, three standard deviations above the mean respectively.

Answer:

Draw a bell - shaped curve. Label the center as 65.5. Label the points on the horizontal axis from left to right as 58, 60.5, 63, 65.5, 68, 70.5, 73 corresponding to three, two, one standard deviations below the mean, the mean, and one, two, three standard deviations above the mean.