QQ. 2 Find probabillties using the normal distribution I QA9
is a normally distributed random variable with mean 5 and standard deviation 3 .
What is the probability that is less than 2 ?
Use the rule and write your answer as a decimal. Round to the nearest
thousandth if necessary.

Ask by Schwartz Davey. in the United States

Jan 24,2025

Question

QQ. 2 Find probabillties using the normal distribution I QA9
is a normally distributed random variable with mean 5 and standard deviation 3 .
What is the probability that is less than 2 ?
Use the rule and write your answer as a decimal. Round to the nearest
thousandth if necessary.

Ask by Schwartz Davey. in the United States

Jan 24,2025

UpStudy AI · Accepted Answer

To find the probability that $$ X $$ is less than 2 for a normally distributed random variable with mean $$ \mu = 5 $$ and standard deviation $$ \sigma = 3 $$, we can use the **68-95-99.7 rule** (also known as the empirical rule).

### Step-by-Step Solution:

1. **Calculate the Z-Score:**
   
   The Z-score helps us determine how many standard deviations away a value is from the mean.
   
   $$
   Z = \frac{X - \mu}{\sigma} = \frac{2 - 5}{3} = \frac{-3}{3} = -1
   $$
   
2. **Apply the 68-95-99.7 Rule:**
   
   - **68%** of the data falls within **±1** standard deviation from the mean.
   - **95%** falls within **±2** standard deviations.
   - **99.7%** falls within **±3** standard deviations.
   
   Since $$ Z = -1 $$ corresponds to one standard deviation below the mean:
   
   - **68%** of the data is between $$ Z = -1 $$ and $$ Z = 1 $$.
   - This leaves **32%** of the data in the two tails, with **16%** in each tail.
   
   Therefore:
   
   $$
   P(X < 2) = P(Z < -1) = 16\% = 0.16
   $$
   
### Final Answer:

The probability that $$ X $$ is less than 2 is **0.160**.
The probability that $$ X $$ is less than 2 is **0.160**.

QQ. 2 Find probabillties using the normal distribution I QA9
is a normally distributed random variable with mean 5 and standard deviation 3 .
What is the probability that is less than 2 ?
Use the rule and write your answer as a decimal. Round to the nearest
thousandth if necessary.

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QQ. 2 Find probabillties using the normal distribution I QA9 is a normally distributed random variable with mean 5 and standard deviation 3 . What is the probability that is less than 2 ? Use the rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

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QQ. 2 Find probabillties using the normal distribution I QA9
is a normally distributed random variable with mean 5 and standard deviation 3 .
What is the probability that is less than 2 ?
Use the rule and write your answer as a decimal. Round to the nearest
thousandth if necessary.