Construct the normal curve from the given µ = 15.55
σ = 0.45

To construct the normal curve with the given mean (µ = 15.55) and standard deviation (σ = 0.45), follow these steps:

1. **Understand the normal distribution:**
   The normal distribution is a symmetric, bell-shaped curve where:
   - The mean (µ) represents the center of the distribution.
   - The standard deviation (σ) controls the spread of the curve. A smaller σ makes the curve narrower, while a larger σ makes the curve wider.

2. **The formula for the normal distribution:**
   The probability density function (PDF) for a normal distribution is given by:

   \[
   f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
   \]

   Substituting the given values:
   - Mean \( \mu = 15.55 \)
   - Standard deviation \( \sigma = 0.45 \)

   The equation becomes:

   \[
   f(x) = \frac{1}{0.45 \sqrt{2\pi}} e^{-\frac{(x-15.55)^2}{2(0.45)^2}}
   \]

3. **Generate the curve:**
   - The peak of the curve occurs at \( x = \mu = 15.55 \).
   - The inflection points (where the curve starts to flatten) occur at \( x = \mu \pm \sigma = 15.55 \pm 0.45 \), i.e., at 15.10 and 16.00.
   - The curve extends infinitely in both directions, but most of the data (99.7%) lies within \( \mu \pm 3\sigma \), or from \( 15.55 - 1.35 = 14.2 \) to \( 15.55 + 1.35 = 16.90 \).

4. **Plotting the curve:**
   - The curve will be centered around 15.55, with a smooth, symmetric decline as you move away from the center.
   - You can plot this curve using software like Python, Excel, or a graphing calculator.

Would you like me to generate this graph or explain more about this process?

Here are five related questions:
1. How do changes in the standard deviation affect the shape of the normal curve?
2. What percentage of data lies within one standard deviation of the mean in a normal distribution?
3. Can we calculate probabilities using this normal curve? How?
4. How does the normal curve differ from other probability distributions, like a uniform or binomial distribution?
5. What is the role of the exponential function in the normal distribution formula?

**Tip:** The area under the normal curve always sums to 1, representing the total probability of all outcomes.

Construct the normal curve from the given µ = 15.55 and σ = 0.45.

Learn how to construct a normal distribution curve using a mean of 15.55 and a standard deviation of 0.45. This example covers key statistical concepts like the probability density function and explains how to plot the curve. Suitable for high school students in Grades 10-12.

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