The accompanying table includes only countries in the top ten in a certain year for both production and consumption of energy.​ (Energy units are quadrillion Btu and population is in​ millions.) What is the​ z-score for Country​ C's population?
Country	Population	Production	Consumption
Country A	1370	101.4	126.2
Country B	327	73.3	99.8
Country C	142	57.7	35.4
Country D	34	15.4	15.9
Country E	1287	13.9	24.7

To find the z-score for Country C's population, we need to follow these steps:

1. **Calculate the mean population** of the given countries.
2. **Calculate the standard deviation** of the populations.
3. Use the formula for the z-score: \[ z = \frac{X - \mu}{\sigma} \]
   where \( X \) is Country C's population, \( \mu \) is the mean population, and \( \sigma \) is the standard deviation.

**Step 1: Calculate the Mean Population**

The populations given are:
- Country A: 1370
- Country B: 327
- Country C: 142
- Country D: 34
- Country E: 1287

The mean population \(\mu\) is calculated as follows:
\[ \mu = \frac{\sum X}{N} = \frac{1370 + 327 + 142 + 34 + 1287}{5} = \frac{3160}{5} = 632 \]

**Step 2: Calculate the Standard Deviation**

First, calculate the variance, which is the average of the squared differences from the mean.
\[ \sigma^2 = \frac{\sum (X_i - \mu)^2}{N} \]

Let's calculate each squared difference:
- For Country A: \((1370 - 632)^2 = 544864\)
- For Country B: \((327 - 632)^2 = 93025\)
- For Country C: \((142 - 632)^2 = 240100\)
- For Country D: \((34 - 632)^2 = 358024\)
- For Country E: \((1287 - 632)^2 = 429529\)

Sum of squared differences:
\[ 544864 + 93025 + 240100 + 358024 + 429529 = 1661542 \]

Variance \(\sigma^2\):
\[ \sigma^2 = \frac{1661542}{5} = 332308.4 \]

Standard deviation \(\sigma\):
\[ \sigma = \sqrt{332308.4} \approx 576.52 \]

**Step 3: Calculate the z-score for Country C's Population**

Now we use the z-score formula:
\[ z = \frac{X - \mu}{\sigma} = \frac{142 - 632}{576.52} = \frac{-490}{576.52} \approx -0.85 \]

So, the z-score for Country C's population is approximately \(-0.85\).

Do you want details or have any questions?

Here are some additional questions you might find interesting:
1. What is the z-score for Country D's population?
2. What is the z-score for Country A's population?
3. How would the mean and standard deviation change if a new country with a population of 500 million was added?
4. Can you calculate the z-score for Country B's energy production?
5. What is the range of the populations in the given countries?

**Tip:** The z-score indicates how many standard deviations an element is from the mean. A negative z-score indicates the value is below the mean, while a positive z-score indicates it is above the mean.

Learn how to calculate the z-score for Country C's population using statistics. Understand the steps involved in finding the z-score, including calculating the mean, standard deviation, and applying the z-score formula. Suitable for advanced high school students and beyond.

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