When Khloe runs the 400 meter dash, her finishing times are normally distributed with a mean of 81 seconds and a standard deviation of 1 second. What percentage of races will her finishing time be slower than 83 seconds, to the nearest tenth?

To find the percentage of races in which Khloe's finishing time will be slower than 83 seconds, we can use the **z-score** formula and then refer to the standard normal distribution table (or use a calculator) to find the corresponding percentage.

### Step 1: Calculate the z-score

The formula for the z-score is:

\[
z = \frac{X - \mu}{\sigma}
\]

Where:
- \( X = 83 \) seconds (the value we're comparing),
- \( \mu = 81 \) seconds (the mean),
- \( \sigma = 1 \) second (the standard deviation).

Substituting the values:

\[
z = \frac{83 - 81}{1} = \frac{2}{1} = 2
\]

### Step 2: Find the percentage corresponding to the z-score

A z-score of 2 corresponds to a cumulative probability (area under the normal curve) of approximately **0.9772**. This means that 97.72% of the times, Khloe will finish **within 83 seconds**.

### Final Answer:

So, **97.7%** (rounded to the nearest tenth) of the time, Khloe's finishing time will be slower than 83 seconds.

---

Would you like more details on z-scores or any other aspect of this problem?

Here are 5 related questions for further exploration:
1. What is the percentage of times Khloe’s finishing time will be slower than 80 seconds?
2. How would the percentage change if the standard deviation were increased to 2 seconds?
3. What is the z-score for a finishing time of 79 seconds?
4. How would you calculate the probability of finishing faster than 78 seconds?
5. What is the probability that Khloe finishes between 80 and 82 seconds?

**Tip:** If you're working with z-scores, you can use online z-score calculators or statistical tables to quickly find corresponding probabilities.

Learn how to calculate the percentage of times Khloe will finish a 400 meter dash slower than 83 seconds, given a normal distribution with a mean of 81 seconds and a standard deviation of 1 second. This problem involves Z-scores and the standard normal distribution, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation