Question
(c) This distribution is relatively symmetric and bell-shaped. Give an interval that is likely to contain about \( 95 \% \) of the data values. Round your answers to two decimal places. The interval is: i eTextbook and Media Save for Later
Ask by Coleman Boyd. in the United States
Feb 03,2025
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Answer
The interval that contains about \( 95\% \) of the data values is from \( \mu - 2\sigma \) to \( \mu + 2\sigma \).
Solution
To find an interval that is likely to contain about \( 95\% \) of the data values for a symmetric and bell-shaped distribution (which typically refers to a normal distribution), we can use the empirical rule (also known as the 68-95-99.7 rule).
According to this rule:
- Approximately \( 68\% \) of the data falls within one standard deviation (\( \sigma \)) of the mean (\( \mu \)).
- Approximately \( 95\% \) of the data falls within two standard deviations of the mean.
Thus, the interval that contains about \( 95\% \) of the data values can be expressed as:
\[
[\mu - 2\sigma, \mu + 2\sigma]
\]
To provide a specific interval, we need the values of the mean (\( \mu \)) and the standard deviation (\( \sigma \)). If you can provide those values, I can calculate the interval for you.
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For a symmetric and bell-shaped distribution, typically the interval that contains about 95% of the data values can be approximated using the empirical rule, also known as the 68-95-99.7 rule. For a standard normal distribution, this will be approximately the mean ± 2 standard deviations. So if your mean is \(\mu\) and standard deviation is \(\sigma\), your interval would be \([\mu - 2\sigma, \mu + 2\sigma]\). Now, if you have your specific values for \(\mu\) and \(\sigma\), just plug them into the formula for your desired interval. Happy calculating!