Question 12 of 16 View Policies Current Attempt in Progress Use the \( 95 \% \) rule and the fact that the summary statistics come from a distribution that is symmetric and bell-shaped to find an interval that is expected to contain about \( 95 \% \) of the data values. A bell-shaped distribution with mean 230 and standard deviation 30 . The interval is \( \mathbf{i} \) to eTextbook and Media
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To find the interval that is expected to contain about \( 95 \% \) of the data values in a symmetric, bell-shaped (normal) distribution, you can use the formula: \( \text{Mean} \pm 2 \times \text{Standard Deviation} \). For this data, the mean is \( 230 \) and the standard deviation is \( 30 \). This gives us: \[ 230 - 2(30) = 170 \] \[ 230 + 2(30) = 290 \] So, the expected interval containing \( 95 \% \) of the data values is from \( 170 \) to \( 290 \). --- The concept of using the \( 95\% \) rule comes from the empirical rule in statistics, which states that in a normal distribution, approximately \( 68\% \) of the data falls within one standard deviation of the mean, \( 95\% \) falls within two standard deviations, and \( 99.7\% \) falls within three standard deviations. This makes it a handy tool for estimating data spread and making predictions based on small sample sizes! When evaluating a bell-shaped distribution, it's important to avoid common mistakes like using the wrong mean or standard deviation values. Double-check your calculations and make sure to visualize the data with a graph if possible. This can help you spot any anomalies, such as outliers, that could skew your interval and cause problems when predicting the distribution of your data.