A standardized​ exam's scores are normally distributed. In a recent​ year, the mean test score was 1495 and the standard deviation was 318. The test scores of four students selected at random are 1910​, 1220​, 2240​, and 1390. Find the​ z-scores that correspond to each value and determine whether any of the values are unusual.
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Part 1
The​ z-score for 1910 is 
  
1.31.
​(Round to two decimal places as​ needed.)
Part 2
The​ z-score for 1220 is 
  
negative 0.86.
​(Round to two decimal places as​ needed.)
Part 3
The​ z-score for 2240 is 
  
2.34.
​(Round to two decimal places as​ needed.)
Part 4
The​ z-score for 1390 is 
  
enter your response here.
​(Round to two decimal places as​ needed.)
The

We can calculate the z-score using the formula:

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

Where:
- \(x\) is the value we are calculating the z-score for,
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.

In this case:
- \(\mu = 1495\),
- \(\sigma = 318\).

### Part 1: \(x = 1910\)
The z-score has already been provided as:
\[
z = \frac{1910 - 1495}{318} = \frac{415}{318} \approx 1.31
\]

### Part 2: \(x = 1220\)
The z-score has already been provided as:
\[
z = \frac{1220 - 1495}{318} = \frac{-275}{318} \approx -0.86
\]

### Part 3: \(x = 2240\)
The z-score has already been provided as:
\[
z = \frac{2240 - 1495}{318} = \frac{745}{318} \approx 2.34
\]

### Part 4: \(x = 1390\)
Now, let's calculate the z-score for the value 1390:
\[
z = \frac{1390 - 1495}{318} = \frac{-105}{318} \approx -0.33
\]

### Interpretation of the z-scores:
- A z-score between -2 and 2 is generally considered normal.
- A z-score beyond ±2 is often considered unusual.

#### Z-scores:
- 1910 → 1.31 (normal),
- 1220 → -0.86 (normal),
- 2240 → 2.34 (unusual),
- 1390 → -0.33 (normal).

Thus, the value 2240 is considered unusual, while the others are within the normal range.

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Do you want more details or have any questions?

Here are 5 related questions to further explore the topic:
1. How would the z-scores change if the mean were 1600?
2. What is the probability of a score being above 1910 based on the z-score?
3. What does a z-score of 0 signify in this context?
4. How do z-scores help in determining the percentile rank of a score?
5. Can a z-score be used to compare scores from different standardized tests?

**Tip:** A z-score tells you how many standard deviations a value is from the mean—positive values are above the mean, and negative values are below the mean.

A standardized​ exam's scores are normally distributed. In a recent​ year, the mean test score was 1495 and the standard deviation was 318. The test scores of four students selected at random are 1910​, 1220​, 2240​, and 1390. Find the​ z-scores that correspond to each value and determine whether any of the values are unusual.

Learn how to calculate z-scores for test scores in a normal distribution with a mean of 1495 and standard deviation of 318. This problem includes test scores of 1910, 1220, 2240, and 1390, and explores whether any scores are considered unusual based on z-scores. Suitable for high school students in Grades 9-12.

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