The expression you're dealing with involves calculating values using the cumulative normal distribution function (NORMSDIST) and the probability density function (NORMDIST). Here is a step-by-step method on how to compute this by hand, or at least break it down for further calculations with a calculator:

Expression:

$ESC = -1,000 \left[ 1 - \text{NORMDIST}\left( \frac{1,000}{707}, 0, 1, 1 \right) \right] + 707 \times \text{NORMDIST}\left( \frac{1,000}{707}, 0, 1, 0 \right) = 25$

Steps:

1. Calculate $\frac{1,000}{707}$:

$\frac{1,000}{707} \approx 1.414$

2. NORMDIST Function Overview:

• NORMDIST(x, mean, standard deviation, cumulative) computes the normal distribution for a given value of $x$, with a given mean and standard deviation.
  • If cumulative = 1, it calculates the cumulative distribution function (CDF).
  • If cumulative = 0, it calculates the probability density function (PDF).

3. Calculate the CDF at $x = 1.414$ (first term):

The cumulative distribution function (CDF) for a standard normal distribution at $x = 1.414$ (mean = 0, standard deviation = 1) can be found using a normal distribution table or an online calculator.

Using a standard normal distribution table: $\text{CDF}(1.414) \approx 0.9214$

So, the first term becomes: $1 - 0.9214 = 0.0786$

4. Calculate the PDF at $x = 1.414$ (second term):

The probability density function (PDF) at $x = 1.414$ (for a standard normal distribution) is given by: $\text{PDF}(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2 / 2}$ Substituting $x = 1.414$: $\text{PDF}(1.414) \approx 0.146$

5. Substitute the values into the equation:

$ESC = -1,000 \times 0.0786 + 707 \times 0.146$

6. Perform the final arithmetic:

$-1,000 \times 0.0786 = -78.6$ $707 \times 0.146 = 103.1$

Thus: $ESC = -78.6 + 103.1 = 24.5 \approx 25$

Final Answer:

$ESC \approx 25$

This matches the result given in the equation. The hand calculation involves using values from the normal distribution table (or a calculator) for both the cumulative distribution function and the probability density function.

Let me know if you want further clarification, and here are five questions that might interest you based on this topic:

1. What is the difference between the cumulative and probability density functions in statistics?
2. How can you approximate values for the CDF without using a table or calculator?
3. How is the normal distribution used in real-world applications?
4. What is the significance of using a standard deviation of 1 in the NORMDIST function?
5. How can you compute the normal distribution for values other than a standard normal distribution?

Tip:

When performing normal distribution calculations by hand, always have access to a standard normal table or an online calculator to get precise values for the CDF and PDF.