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The question asks to find the z-score of the median from a normally distributed population with a sample of 12 data points. Let's break this down step by step:

### Step 1: Arrange the data in ascending order
Here are the 12 data points given:
\[ 46, 47, 51, 52, 55, 56, 67, 68, 70, 71, 71, 78 \]

These values are already ordered, so no need for further sorting.

### Step 2: Calculate the median
Since there are 12 values, the median will be the average of the 6th and 7th data points (since the number of observations is even).

Median = \( \frac{56 + 67}{2} = \frac{123}{2} = 61.5 \)

### Step 3: Find the mean and standard deviation
Now, we calculate the sample mean and standard deviation.

#### Mean (\( \mu \)):
\[
\mu = \frac{46 + 47 + 51 + 52 + 55 + 56 + 67 + 68 + 70 + 71 + 71 + 78}{12}
\]
Let me calculate this.The mean of the data set is \( \mu = 61.0 \).

### Step 4: Calculate the standard deviation (\( \sigma \))
Now let's compute the sample standard deviation.The sample standard deviation is approximately \( \sigma = 10.95 \).

### Step 5: Calculate the z-score of the median
The formula for the z-score is:

\[
z = \frac{X - \mu}{\sigma}
\]

Where:
- \( X \) is the median, \( 61.5 \),
- \( \mu \) is the mean, \( 61.0 \),
- \( \sigma \) is the standard deviation, \( 10.95 \).

Let's calculate the z-score.The z-score of the median is approximately \( 0.046 \).

Would you like further clarification on any part of this solution? Here are a few related questions that may interest you:

1. How would the z-score change if one of the data points is altered?
2. What is the significance of a z-score in a normally distributed population?
3. Can the z-score be calculated directly from the data set without computing the median?
4. How is the z-score applied in hypothesis testing?
5. What if the population was skewed instead of normally distributed?

**Tip:** The z-score helps identify how far a particular data point (or statistic like the median) is from the mean in terms of standard deviations.

A random experiment involves drawing a sample of 12 data values from a normally distributed population. The random variable is the z-score of the median of the data set. The data values are: [46, 47, 51, 52, 55, 56, 67, 68, 70, 71, 71, 78]. Calculate the z-score of the median.

Learn how to calculate the z-score of the median for a sample of 12 data points drawn from a normally distributed population. This problem involves calculating the median, mean, and standard deviation, and applying the z-score formula to the median. Suitable for college-level statistics students.

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