When we combine random variables that each follow a normal distribution, the resulting distribution is also normally distributed. This lets us answer questions about the resulting distribution.

## Example Problems: Combining Normal Random Variables

**Example 1**

**Andy, Bob, Craig, and Doug play a round of golf together each week. Each of their scores are normally distributed with a mean of 70 strokes and a standard deviation of 5 strokes. ****Find the probability that the sum of their scores in any given week is higher than 300 strokes.**

**Step 1: Find the mean number of total strokes.**

μ_{total} = μ_{Andy} + μ_{Bob }+ μ_{Craig }+ μ_{Doug}

μ_{total} = 70 + 70 + 70 +70 = **280**

**Step 2: Find the standard deviation of total strokes.**

σ^{2}_{total} = σ^{2}_{Andy} + σ^{2}_{Bob }+ σ^{2}_{Craig }+ σ^{2}_{Doug}

σ^{2}_{total} = 5^{2} + 5^{2}_{ }+ 5^{2}_{ }+ 5^{2} = 100

σ_{total} = √100 = **10**

**Step 3: Use the Z Score Area Calculator to find the probability that the sum of their scores in a given week is higher than 300 strokes. **

The probability that the sum of their scores in a given week is above 300 is **0.02275**.

**Example 2**

*Meredith and Angela are both saleswomen at a paper company. Meredith’s weekly sales are normally distributed with a mean of 50 sales and a standard deviation of 2 sales. Angela’s weekly sales are normally distributed with a mean of 45 sales and a standard deviation of 2 sales.*

*Assume that their sales in any given week are independent. **Find the probability that Angela has more sales than Meredith in a randomly selected week.*

**Step 1: Find the mean of the difference in sales.**

μ_{diff} = μ_{Meredith} – μ_{Angela}

μ_{diff} = 50 – 45 = **5**

**Step 2: Find the standard deviation of the difference in sales.**

σ^{2}_{diff} = σ^{2}_{Meredith} + σ^{2}_{Angela}

σ^{2}_{diff} = 2^{2} + 2^{2} = 4

σ_{diff} = √4 = **2**

**Step 3: Use the Z Score Area Calculator to find the probability that the difference between Meredith and Angela’s sales is less than zero in a given week.**

The probability that the difference between Meredith and Angela’s sales is less than zero in a given week. (i.e. Angela sells more) is** 0.00621.**

**Example 3**

**Mike, Greg, Austin, and Tony play a game of bowling each week together. Each of their scores are normally distributed with a mean of 100 and a standard deviation of 5. ****Find the probability that the sum of their scores in any given week is lower than 415.**

**Step 1: Find the mean number of total strokes.**

μ_{total} = μ_{Mike} + μ_{Greg }+ μ_{Austin }+ μ_{Tony}

μ_{total} = 100 + 100 + 100 + 100 = **400**

**Step 2: Find the standard deviation of total strokes.**

σ^{2}_{total} = σ^{2}_{Mike} + σ^{2}_{Greg }+ σ^{2}_{Austin }+ σ^{2}_{Tony}

σ^{2}_{total} = 5^{2} + 5^{2}_{ }+ 5^{2}_{ }+ 5^{2} = 100

σ_{total} = √100 = **10**

**Step 3: Use the Z Score Area Calculator to find the probability that the sum of their scores in a given week is lower than 415. **

The probability that the sum of their scores in a given week is below 415 is **0.93319**.

**Example 4**

*Cody and Kyle are both salesmen at a sneaker company. Cody’s weekly sales are normally distributed with a mean of 30 sales and a standard deviation of 2 sales. Kyle’s weekly sales are normally distributed with a mean of 25 sales and a standard deviation of 2 sales.*

*Assume that their sales in any given week are independent. Find the probability that Kyle has more sales than Cody in a randomly selected week.*

**Step 1: Find the mean of the difference in sales.**

μ_{diff} = μ_{Cody} – μ_{Kyle}

μ_{diff} = 30 – 25 = **5**

**Step 2: Find the standard deviation of the difference in sales.**

σ^{2}_{diff} = σ^{2}_{Cody} + σ^{2}_{Kyle}

σ^{2}_{diff} = 2^{2} + 2^{2} = 4

σ_{diff} = √4 = **2**

**Step 3: Use the Z Score Area Calculator to find the probability that the difference between Cody and Kyle’s sales is less than zero in a given week.**

The probability that the difference between Cody and Kyle’s sales is less than zero in a given week. (i.e. Kyle sells more) is** 0.00621.**