Consider a sample with six observations of 16, 11, 13, 22, 15, and 19. Compute the z-score for each observation. (Leave no cells blank - be certain to enter "0" wherever required. Round your answers to 2 decimal places. Negative values should be indicated by a minus sign.)
 




To compute the z-score for each observation in the sample, we use the formula:

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

Where:
- \(x\) is the observation,
- \(\mu\) is the mean of the sample,
- \(\sigma\) is the standard deviation of the sample.

### Step 1: Compute the mean (\(\mu\)) of the sample
The sample is \(16, 11, 13, 22, 15, 19\).

\[
\mu = \frac{16 + 11 + 13 + 22 + 15 + 19}{6} = \frac{96}{6} = 16
\]

### Step 2: Compute the standard deviation (\(\sigma\))
The standard deviation is computed using the formula for the sample standard deviation:

\[
\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n-1}}
\]

Where:
- \(x_i\) is each observation,
- \(\mu\) is the mean,
- \(n\) is the number of observations.

Let's calculate it:

1. \((16 - 16)^2 = 0\)
2. \((11 - 16)^2 = 25\)
3. \((13 - 16)^2 = 9\)
4. \((22 - 16)^2 = 36\)
5. \((15 - 16)^2 = 1\)
6. \((19 - 16)^2 = 9\)

Summing these squared differences:

\[
0 + 25 + 9 + 36 + 1 + 9 = 80
\]

Now, compute the standard deviation:

\[
\sigma = \sqrt{\frac{80}{6-1}} = \sqrt{\frac{80}{5}} = \sqrt{16} = 4
\]

### Step 3: Compute the z-scores
Now that we have \(\mu = 16\) and \(\sigma = 4\), we can compute the z-score for each observation:

1. For \(x = 16\):
   \[
   z = \frac{16 - 16}{4} = 0
   \]

2. For \(x = 11\):
   \[
   z = \frac{11 - 16}{4} = \frac{-5}{4} = -1.25
   \]

3. For \(x = 13\):
   \[
   z = \frac{13 - 16}{4} = \frac{-3}{4} = -0.75
   \]

4. For \(x = 22\):
   \[
   z = \frac{22 - 16}{4} = \frac{6}{4} = 1.50
   \]

5. For \(x = 15\):
   \[
   z = \frac{15 - 16}{4} = \frac{-1}{4} = -0.25
   \]

6. For \(x = 19\):
   \[
   z = \frac{19 - 16}{4} = \frac{3}{4} = 0.75
   \]

### Final z-scores:
- For \(x = 16\), z-score = 0
- For \(x = 11\), z-score = -1.25
- For \(x = 13\), z-score = -0.75
- For \(x = 22\), z-score = 1.50
- For \(x = 15\), z-score = -0.25
- For \(x = 19\), z-score = 0.75

Would you like any more details or have any further questions?

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Here are 5 related questions to deepen your understanding:
1. What is the significance of a z-score in statistics?
2. How does changing the sample size affect the standard deviation?
3. Can z-scores be used to compare values from different datasets?
4. What are the properties of a normal distribution in terms of z-scores?
5. How does skewness in a dataset affect the z-scores?

**Tip:** A z-score helps determine how far a data point is from the mean in units of standard deviation.

Consider a sample with six observations of 16, 11, 13, 22, 15, and 19. Compute the z-score for each observation. (Leave no cells blank - be certain to enter '0' wherever required. Round your answers to 2 decimal places. Negative values should be indicated by a minus sign.)

Learn how to compute z-scores for a dataset with six observations: 16, 11, 13, 22, 15, and 19. This problem involves key statistical concepts like mean, standard deviation, and z-scores, and is suitable for high school students in Grades 9-12.

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