The questions from the image are related to the properties of a normal distribution. Let’s solve them systematically:

The salaries of employees follow a normal distribution with:

• Mean (μ) = €25,000
• Standard deviation (σ) = €5,000

A. Probability of a salary of at least €31,000

To find the probability, we calculate the z-score: $z = \frac{x - \mu}{\sigma}$ Here, $x = 31,000$, $\mu = 25,000$, and $\sigma = 5,000$: $z = \frac{31,000 - 25,000}{5,000} = \frac{6,000}{5,000} = 1.2$

Using the standard normal distribution table, the cumulative probability for $z = 1.2$ is approximately 0.8849.

The probability of $x \geq 31,000$ is: $P(x \geq 31,000) = 1 - P(z \leq 1.2) = 1 - 0.8849 = 0.1151$

Answer: The probability is 11.51%.

B. Percentage of employees earning less than €12,200

Similarly, calculate the z-score for $x = 12,200$: $z = \frac{12,200 - 25,000}{5,000} = \frac{-12,800}{5,000} = -2.56$

Using the z-table, the cumulative probability for $z = -2.56$ is approximately 0.0052.

Answer: The percentage is 0.52%.

C. Minimum and maximum salaries for the middle 95%

For the middle 95% of a normal distribution, the z-scores are approximately ±1.96.

The corresponding values are calculated as: $x = \mu + z \cdot \sigma$ For the minimum: $x_{\text{min}} = 25,000 + (-1.96) \cdot 5,000 = 25,000 - 9,800 = 15,200$ For the maximum: $x_{\text{max}} = 25,000 + (1.96) \cdot 5,000 = 25,000 + 9,800 = 34,800$

Answer: The salaries range from €15,200 to €34,800.

D. Percentage of employees earning between €20,000 and €35,000

Calculate the z-scores for both values:

1. For $x = 20,000$: $z = \frac{20,000 - 25,000}{5,000} = \frac{-5,000}{5,000} = -1$ The cumulative probability for $z = -1$ is approximately 0.1587.

2. For $x = 35,000$: $z = \frac{35,000 - 25,000}{5,000} = \frac{10,000}{5,000} = 2$ The cumulative probability for $z = 2$ is approximately 0.9772.

The probability of being between $x = 20,000$ and $x = 35,000$ is: $P(20,000 \leq x \leq 35,000) = P(z \leq 2) - P(z \leq -1) = 0.9772 - 0.1587 = 0.8185$

Answer: The percentage is 81.85%.

Any Questions or Clarifications?

Would you like detailed calculations or more context?

5 Related Questions:

1. What is the probability of earning between €15,000 and €30,000?
2. How does the standard deviation affect the probability calculations?
3. What percentage of employees earn above €40,000?
4. How would the range for the middle 99% of salaries differ from the 95%?
5. Can a salary of €50,000 be considered an outlier in this distribution?

Tip:

The z-score is key in solving normal distribution problems. Memorizing common z-values (e.g., ±1.96 for 95%) can save time!