To calculate the probability of $3 \leq x \leq 8$ for a normally distributed variable $x$ with a mean of 8 and a standard deviation of 4 in Excel, follow these steps:

Steps:

1. Open Excel and choose a cell to start your calculations.

2. Use the NORM.DIST function to calculate cumulative probabilities. The general format is:

   =NORM.DIST(x, mean, standard_deviation, cumulative)
   

   • $x$ is the value of interest (3 or 8).
   • The mean is 8.
   • The standard deviation is 4.
   • For cumulative distribution, set cumulative to TRUE.

3. Calculate the cumulative probability for $x = 8$:

   In a chosen cell, enter:

   =NORM.DIST(8, 8, 4, TRUE)
   

4. Calculate the cumulative probability for $x = 3$:

   In another cell, enter:

   =NORM.DIST(3, 8, 4, TRUE)
   

5. Find the probability for the range $3 \leq x \leq 8$:

   Subtract the cumulative probability for $x = 3$ from the cumulative probability for $x = 8$:

   =NORM.DIST(8, 8, 4, TRUE) - NORM.DIST(3, 8, 4, TRUE)
   

   This will give you the desired probability for $3 \leq x \leq 8$.

Example Calculation in Excel:

• Probability for $x = 8$:
  =NORM.DIST(8, 8, 4, TRUE)
  Result ≈ 0.5

• Probability for $x = 3$:
  =NORM.DIST(3, 8, 4, TRUE)
  Result ≈ 0.1056

• Final probability for $3 \leq x \leq 8$:
  = 0.5 - 0.1056
  Result ≈ 0.3944

This means that the probability that $x$ lies between 3 and 8 is approximately 0.3944 or 39.44%.

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Do you want any additional details or have further questions?

Here are 5 related questions for practice:

1. How do you calculate the probability of $x \geq 10$ for the same distribution?
2. How would you calculate $P(x \leq 5)$ using Excel?
3. What is the probability that $x$ lies outside the range 3 and 8?
4. How do you interpret the result from NORM.DIST when cumulative is set to FALSE?
5. How can you find the z-score for $x = 6$ using Excel?

Tip: Excel’s NORM.S.DIST function can be used for the standard normal distribution (mean = 0, standard deviation = 1).