Questions: A set of data items is normally distributed with a mean of 700 and a standard deviation of 10. Find the data item in this distribution that corresponds to the given z-score. z=-6 The data item that corresponds to z=-6 is (Type an integer or a decimal.)

A set of data items is normally distributed with a mean of 700 and a standard deviation of 10. Find the data item in this distribution that corresponds to the given z-score. z=-6 The data item that corresponds to z=-6 is (Type an integer or a decimal.)
Transcript text: A set of data items is normally distributed with a mean of 700 and a standard deviation of 10 . Find the data item in this distribution that corresponds to the given z-score. \[ z=-6 \] The data item that corresponds to $z=-6$ is $\square$ (Type an integer or a decimal.)
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Solution

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Solution Steps

Step 1: Given Information

We have a normally distributed dataset with the following parameters:

  • Mean (\( \mu \)) = 700
  • Standard Deviation (\( \sigma \)) = 10
  • Z-score (\( z \)) = -6
Step 2: Z-score Formula

To find the data item corresponding to the given z-score, we use the formula: \[ X = \mu + z \cdot \sigma \] where:

  • \( X \) is the data item,
  • \( \mu \) is the mean,
  • \( z \) is the z-score,
  • \( \sigma \) is the standard deviation.
Step 3: Calculation

Substituting the known values into the formula: \[ X = 700 + (-6) \cdot 10 \] \[ X = 700 - 60 \] \[ X = 640 \]

Final Answer

The data item that corresponds to \( z = -6 \) is \(\boxed{640}\).

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