This question has two parts. First, answer Part A. Then, answer Part B.
Part A
STATISTICS The most familiar statistical measure is the arithmetic mean, or average. A second important statistical measure is the standard deviation, which is a measure of how far the data are from the mean. For example, the mean score on the Wechsler IQ test is 100 and the standard deviation is 15. This means that people within one standard deviation of the mean have IQ scores that are 15 points higher or lower than the mean.

 

a. One year, the mean mathematics score on the ACT test was 20.9 with a standard deviation of 5.3. Write an absolute value equation to find the maximum and minimum scores within one standard deviation of the mean. Let t represent the test score.

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Part B
Fill in the blank question.
b. What is the range of ACT mathematics scores within one standard deviation of the mean? List the scores from least to greatest.

 



 

What is the range of ACT mathematics scores within two standard deviations of the mean? List the scores from least to greatest.

### Part A:
We are asked to write an absolute value equation that represents scores within one standard deviation of the mean on the ACT mathematics test. 

- **Mean score (µ)** = 20.9
- **Standard deviation (σ)** = 5.3

Within one standard deviation, the scores will range from \( \mu - \sigma \) to \( \mu + \sigma \).

Thus, we want to find \( t \) (the test score) such that it lies within one standard deviation of the mean. This is modeled by the following absolute value equation:
\[
|t - 20.9| = 5.3
\]
This equation expresses that the distance between the test score \( t \) and the mean (20.9) is 5.3 points, which is exactly one standard deviation.

---

### Part B:
Now, we will find the range of ACT mathematics scores within **one** and **two** standard deviations of the mean.

#### 1. **One standard deviation range:**
The range is from:
\[
\text{Minimum score} = 20.9 - 5.3 = 15.6
\]
\[
\text{Maximum score} = 20.9 + 5.3 = 26.2
\]
So, the range of scores within one standard deviation is **15.6 to 26.2**.

#### 2. **Two standard deviations range:**
Now, we consider scores within two standard deviations:
\[
\text{Minimum score} = 20.9 - 2(5.3) = 20.9 - 10.6 = 10.3
\]
\[
\text{Maximum score} = 20.9 + 2(5.3) = 20.9 + 10.6 = 31.5
\]
So, the range of scores within two standard deviations is **10.3 to 31.5**.

---

### Summary:
- **One standard deviation range:** 15.6 to 26.2
- **Two standard deviations range:** 10.3 to 31.5

Would you like further details on standard deviation, or any other clarification?

### Related Questions:
1. What is the formula for calculating the standard deviation?
2. How do you interpret data points that are beyond two standard deviations from the mean?
3. Can there be negative values for test scores in such distributions?
4. How do we calculate the z-score for any given data point?
5. Why is the standard deviation important in statistics?

### Tip:
The wider the range of standard deviations, the more data points (or test scores, in this case) you include in your analysis. Typically, about 68% of data falls within one standard deviation, while about 95% falls within two.

This question has two parts. First, answer Part A. Then, answer Part B.
Part A
STATISTICS The most familiar statistical measure is the arithmetic mean, or average. A second important statistical measure is the standard deviation, which is a measure of how far the data are from the mean. For example, the mean score on the Wechsler IQ test is 100 and the standard deviation is 15. This means that people within one standard deviation of the mean have IQ scores that are 15 points higher or lower than the mean.

 a. One year, the mean mathematics score on the ACT test was 20.9 with a standard deviation of 5.3. Write an absolute value equation to find the maximum and minimum scores within one standard deviation of the mean. Let t represent the test score.

Part B
Fill in the blank question.
b. What is the range of ACT mathematics scores within one standard deviation of the mean? List the scores from least to greatest.

What is the range of ACT mathematics scores within two standard deviations of the mean? List the scores from least to greatest.

Learn how to calculate the ACT mathematics test score range within one and two standard deviations from the mean of 20.9 with a standard deviation of 5.3. This problem covers key statistical concepts such as mean, standard deviation, and absolute value equations. Suitable for students in Grades 9-11.

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