Suppose that the distribution of scores on a statewide standardized test are bell-shaped with a mean of 74 and a standard deviation of 3. Estimate the percentage of scores that were...

(a) ...between 65 and 83.

%

(b) ...above 80.

%

(c) ...below 65.

%

(d) ...between 71 and 83.

%

We are dealing with a bell-shaped distribution, which typically follows the empirical rule (also known as the 68-95-99.7 rule) for a normal distribution. According to this rule:

- About **68%** of the data falls within **1 standard deviation** of the mean.
- About **95%** of the data falls within **2 standard deviations** of the mean.
- About **99.7%** of the data falls within **3 standard deviations** of the mean.

Given:
- Mean \( \mu = 74 \)
- Standard deviation \( \sigma = 3 \)

Let's break this down for each part:

### (a) Between 65 and 83

1. **65** is \( 74 - 3 \times 3 = 74 - 9 = 65 \), which is **3 standard deviations** below the mean.
2. **83** is \( 74 + 3 \times 3 = 74 + 9 = 83 \), which is **3 standard deviations** above the mean.

Since the range from 65 to 83 covers from **3 standard deviations below** to **3 standard deviations above** the mean, we know from the empirical rule that **99.7%** of the data falls within 3 standard deviations.

Thus, approximately **99.7%** of the scores are between 65 and 83.

### (b) Above 80

1. **80** is \( \frac{80 - 74}{3} = 2 \) standard deviations **above** the mean.

From the empirical rule, we know that about **95%** of the data lies within **2 standard deviations** of the mean (i.e., between 68 and 80). This leaves **5%** outside of this range, with half of that (i.e., **2.5%**) above 80.

Thus, about **2.5%** of the scores are above 80.

### (c) Below 65

As noted in part (a), **65** is **3 standard deviations below** the mean. From the empirical rule, **0.15%** of the data lies beyond 3 standard deviations on either side.

Thus, approximately **0.15%** of the scores are below 65.

### (d) Between 71 and 83

1. **71** is \( \frac{71 - 74}{3} = -1 \) standard deviation **below** the mean.
2. **83** is \( \frac{83 - 74}{3} = 3 \) standard deviations **above** the mean.

From the empirical rule:
- **68%** of the data is within **1 standard deviation** of the mean (i.e., between 71 and 77).
- **99.7%** of the data is within **3 standard deviations** of the mean (i.e., between 65 and 83).

Thus, the percentage of scores between 71 and 83 is the sum of the percentages from **1 standard deviation below** and up to **3 standard deviations above**:
- Half of **68%** (from 71 to 74) = **34%**
- The remainder from 74 to 83 is half of **99.7%** = **49.85%**.

Summing these gives approximately **83.85%** of the scores between 71 and 83.

### Summary:
- (a) Between 65 and 83: **99.7%**
- (b) Above 80: **2.5%**
- (c) Below 65: **0.15%**
- (d) Between 71 and 83: **83.85%**

---

Would you like further clarification or details on these calculations?

Here are some related questions:
1. How would the percentages change if the standard deviation were larger?
2. What percentage of scores would be between 74 and 80?
3. Can you find the z-scores for values of 68 and 77?
4. What is the probability of getting a score above 83?
5. How does the empirical rule differ from using exact probabilities with the z-table?

**Tip:** When working with normal distributions, always try to visualize the bell curve to better understand the distribution of data points.

Suppose that the distribution of scores on a statewide standardized test are bell-shaped with a mean of 74 and a standard deviation of 3. Estimate the percentage of scores that were...

(a) ...between 65 and 83.
(b) ...above 80.
(c) ...below 65.
(d) ...between 71 and 83.

Explore how to estimate the percentage of scores in a normal distribution with a mean of 74 and standard deviation of 3. Using the empirical rule, learn to calculate the percentage of scores between 65 and 83, above 80, below 65, and between 71 and 83.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation