A set of data items is normally distributed with a mean of 800 and a standard deviation of 20. Find the data item in this distribution that corresponds to the given \( z \)-score. \( z=7 \) The data item that corresponds to \( z=7 \) is \( \square \). (Type an integer or a decimal.)

To find the data item corresponding to the \( z \)-score, you can use the formula: \[ X = \mu + z \cdot \sigma \] Where \( \mu \) is the mean (800), \( z \) is the \( z \)-score (7), and \( \sigma \) is the standard deviation (20). Plugging in the values: \[ X = 800 + 7 \cdot 20 = 800 + 140 = 940 \] So, the data item that corresponds to \( z=7 \) is \( 940 \). Evaluating a \( z \)-score of 7 is quite rare in practical scenarios, making this distribution analysis particularly interesting! Most data in a normal distribution lie within 3 standard deviations of the mean, so stepping outside gives you a clearer view of outliers or extreme values in your data set. When dealing with \( z \)-scores, keep in mind the significance of values that fall far from the mean. Common mistakes include miscalculating the formula or not properly interpreting the \( z \)-score, leading to confusion about how the data compares to its mean and standard deviation.