Solving Problems Using the Normal Distribution Revision Notes for VCE SSCE Mathematical Methods

Solving Problems Using the Normal Distribution

Introduction to practical applications

The normal distribution is a powerful tool that can be applied to solve many real-world problems. When working with normally distributed data, we can calculate probabilities and determine unknown parameters using the properties of the normal curve.

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The normal distribution's practical applications extend across numerous fields including quality control, psychology, medicine, and finance. Understanding how to work with normally distributed data is an essential skill for statistical analysis.

In practical situations, you'll often need to:

Find probabilities for given values
Calculate conditional probabilities
Determine the mean and standard deviation when these parameters are unknown

Using technology to find probabilities

Calculator methods

Modern calculators provide functions specifically designed for normal distribution calculations. The Normal CDF (Cumulative Distribution Function) allows you to find the probability that a normally distributed variable falls within a certain range.

TI-Nspire Calculator:

Navigate to menu > Probability > Distributions > Normal Cdf
Enter the lower bound (use $-\infty$ for no lower limit)
Enter the upper bound
Specify the mean ( $\mu$ ) and standard deviation ( $\sigma$ )
The calculator returns the probability

Casio ClassPad:

Access Interactive > Distribution > Continuous > normCDf
Input the lower bound, upper bound, standard deviation, and mean
The calculator displays the probability
You can also view a graphical representation of the shaded area

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Critical Calculator Tips:

Always check that you've entered the parameters in the correct order - different calculators may require different input sequences
Use $-\infty$ or a very large negative number (e.g., -99999) for no lower bound
Use $\infty$ or a very large positive number (e.g., 99999) for no upper bound
Verify your result makes sense: all probabilities must be between 0 and 1

Worked example: calculating basic probabilities

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Worked Example: Basic Probability Calculation

The time taken to complete a logical reasoning task is normally distributed with a mean of 55 seconds and a standard deviation of 8 seconds.

Part a) Find the probability that a randomly chosen person takes less than 50 seconds.

Solution:

Let $X$ represent the completion time in seconds, where $X \sim N(55, 8^2)$ .

We need to find $\text{Pr}(X < 50)$ .

Using the Normal CDF function with:

Lower bound: $-\infty$
Upper bound: $50$
Mean ( $\mu$ ): $55$
Standard deviation ( $\sigma$ ): $8$

The calculator gives:

\text{Pr}(X < 50) = 0.2660

(to 4 decimal places)

This means approximately 26.6% of people will complete the task in less than 50 seconds.

Conditional probability with normal distributions

Conditional probability involves finding the probability of an event occurring given that another event has already occurred. With normal distributions, we use the formula:

\text{Pr}(A|B) = \frac{\text{Pr}(A \cap B)}{\text{Pr}(B)}

When working with inequalities involving the same variable, this simplifies because the intersection represents the more restrictive condition.

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Understanding the Intersection: When dealing with two conditions like $X < a$ and $X < b$ where $a < b$ , the intersection $X < a \cap X < b$ simply equals $X < a$ because any value less than $a$ is automatically less than $b$ . Always identify which condition is more restrictive!

Applying conditional probability

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Worked Example: Conditional Probability

Part b) Find the probability that a randomly chosen person takes less than 50 seconds, given that they took less than 60 seconds.

Solution:

We need to find $\text{Pr}(X < 50 | X < 60)$ .

Using the conditional probability formula:

\text{Pr}(X < 50 | X < 60) = \frac{\text{Pr}(X < 50 \cap X < 60)}{\text{Pr}(X < 60)}

Since $X < 50$ is more restrictive than $X < 60$ , the intersection simply equals $\text{Pr}(X < 50)$ :

\text{Pr}(X < 50 | X < 60) = \frac{\text{Pr}(X < 50)}{\text{Pr}(X < 60)}

First, find $\text{Pr}(X < 60)$ using Normal CDF:

\text{Pr}(X < 60) = 0.7340

Now substitute both probabilities:

\text{Pr}(X < 50 | X < 60) = \frac{0.2660}{0.7340}

= 0.3624

(to 4 decimal places)

This result means that if we know someone completed the task in under 60 seconds, there's a 36.24% chance they did so in under 50 seconds.

Finding unknown parameters

Sometimes the mean and standard deviation of a normal distribution are unknown. In these cases, we can use given probability information to determine these parameters. This often requires:

Transforming to the standard normal distribution $Z \sim N(0,1)$
Using the inverse normal function to find z-scores
Applying the symmetry property of the normal curve

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The process of finding unknown parameters is sometimes called "reverse engineering" the normal distribution. Instead of using known parameters to find probabilities, we use known probabilities to find the parameters. This technique is particularly useful in quality control and manufacturing processes.

Transformation formula

To transform a normal variable $X \sim N(\mu, \sigma^2)$ to the standard normal variable $Z$ :

Z = \frac{X - \mu}{\sigma}

This standardization process allows us to work with the standard normal distribution, which has well-documented properties and is built into calculator functions.

Worked example with unknown parameters

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Worked Example: Finding Unknown Mean and Standard Deviation

Metal rods have lengths that are normally distributed. Quality control limits are set at 1.925 cm and 2.075 cm. Observations show that 5% of rods are rejected as undersized and 5% as oversized.

Find the mean and standard deviation of the distribution.

Solution:

From the given information:

$\text{Pr}(X > 2.075) = 0.05$
$\text{Pr}(X < 1.925) = 0.05$

Step 1: Find the mean using symmetry

The normal distribution is symmetrical. Since 5% lie above 2.075 and 5% lie below 1.925, the mean must be exactly halfway between these values:

\mu = \frac{2.075 + 1.925}{2} = \frac{4.000}{2} = 2 \text{ cm}

Step 2: Transform to standard normal

Transform the given probabilities:

\text{Pr}\left(Z > \frac{2.075 - \mu}{\sigma}\right) = 0.05

and

\text{Pr}\left(Z < \frac{1.925 - \mu}{\sigma}\right) = 0.05

Step 3: Rewrite using complements

The first probability can be rewritten as:

\text{Pr}\left(Z < \frac{2.075 - \mu}{\sigma}\right) = 0.95

Step 4: Use inverse normal

Using the inverse normal function on a calculator:

\frac{2.075 - \mu}{\sigma} = 1.6448...

and

\frac{1.925 - \mu}{\sigma} = -1.6448...

These equations confirm our mean value $\mu = 2$ (notice the symmetry in the z-scores).

Step 5: Solve for standard deviation

Substitute $\mu = 2$ into the first equation:

\frac{2.075 - 2}{\sigma} = 1.6448...

\frac{0.075}{\sigma} = 1.6448...

\sigma = \frac{0.075}{1.6448...}

\sigma = 0.045596...

Therefore $\sigma = 0.0456$ cm (to 4 decimal places)

Answer: The mean length is 2 cm and the standard deviation is 0.0456 cm.

Key strategies for problem solving

When solving problems involving the normal distribution:

Identify the parameters: Clearly state the mean ( $\mu$ ) and standard deviation ( $\sigma$ ) if known
Use calculator technology: Leverage Normal CDF for finding probabilities efficiently
Apply conditional probability carefully: Remember that $\text{Pr}(A|B) = \frac{\text{Pr}(A \cap B)}{\text{Pr}(B)}$
Use symmetry: The normal distribution is symmetric about its mean
Transform when needed: Convert to standard normal when parameters are unknown
Check your working: Ensure probabilities are between 0 and 1, and that your answers make practical sense

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Common Pitfalls to Avoid:

Confusing variance ( $\sigma^2$ ) with standard deviation ( $\sigma$ )
Forgetting to check that the more restrictive condition in conditional probability problems
Not utilizing the symmetry property when it could simplify calculations
Entering calculator values in the wrong order

Remember!

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Key Points to Remember:

The Normal CDF function on your calculator is the most efficient tool for finding probabilities in normally distributed data
For conditional probability with normal distributions, use:

\text{Pr}(X < a | X < b) = \frac{\text{Pr}(X < a)}{\text{Pr}(X < b)}

when $a < b$

The symmetry of the normal distribution is powerful: if equal percentages are rejected at both tails, the mean is exactly halfway between the rejection limits
Transform to the standard normal distribution using:

Z = \frac{X - \mu}{\sigma}

when you need to find unknown parameters

The inverse normal function helps you find the value corresponding to a given probability, which is essential when determining unknown means and standard deviations

Solving Problems Using the Normal Distribution (VCE SSCE Mathematical Methods): Revision Notes