Heights of women have a bell-shaped distribution with a mean of 158 cm and a standard deviation of 8 cm. Using Chebyshev’s theorem, what do we know about the percentage of Women with heights that are within 3 standard deviations of the mean. What are the minimum and maximum heights that are within 3 standard deviations of the mean
This is like the eighth statistical problem I have put up, I am trying to get some help to learn how to write these problems out. This is my last class, Please can some one help me here!!
Z = (score-mean)/SD
±3 = (score-158)/8
Solve for both the negative and positive values.
Sure! I'd be happy to help you with your statistical problem. Let's break it down step by step:
1. Chebyshev's theorem: Chebyshev's theorem states that for any dataset (regardless of shape), at least (1 - 1/k^2) percent of the data falls within k standard deviations of the mean, where k is any positive number greater than 1.
2. In this case, we know that the heights of women have a bell-shaped distribution (which indicates a normal distribution) with a mean of 158 cm and a standard deviation of 8 cm.
3. To find the percentage of women with heights within 3 standard deviations of the mean using Chebyshev's theorem, we need to find k. In this case, k is equal to 3.
4. According to Chebyshev's theorem, at least (1 - 1/k^2) percent of the data falls within k standard deviations of the mean. Therefore, at least (1 - 1/3^2) = 1 - 1/9 = 8/9 (approximately 0.889, or 88.9%) of the women's heights fall within 3 standard deviations of the mean.
5. Next, we'll calculate the minimum and maximum heights that are within 3 standard deviations of the mean. To do this, we'll use the formula:
- Minimum height = mean - (k * standard deviation)
- Maximum height = mean + (k * standard deviation)
6. Plugging in the given values, we have:
- Minimum height = 158 cm - (3 * 8 cm) = 158 cm - 24 cm = 134 cm
- Maximum height = 158 cm + (3 * 8 cm) = 158 cm + 24 cm = 182 cm
Therefore, we know that at least 88.9% of the women's heights fall within 3 standard deviations of the mean (between 134 cm and 182 cm, inclusive).