A survey tracks a nationally representative sample of students from eighth grade through high school and college. Research published in a journal examined the scores of 250 students who paid a private tutor to help them improve their scores. The table below summarizes the changes in both the Mathematics and Verbal scores for these students.
MATH	VERBAL
Mean change in score	19	14
Standard deviation of score changes	65	50
 
Step 3 of 3 :  Suppose one of the 250 students who paid a private tutor is selected at random. Suppose the selected​ student's score increased on one of the tests by 188 points. Which​ test, the Math or​ Verbal, is the one most likely to have had the 188​-point ​increase?

To determine which test is more likely to have the 188-point increase, we can calculate the **z-score** for the increase in both Math and Verbal scores. The z-score helps us standardize the change by comparing it to the average change and standard deviation.

The formula for the z-score is:

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

Where:
- \(x\) is the observed value (188 in this case),
- \(\mu\) is the mean change in score,
- \(\sigma\) is the standard deviation of score changes.

### Step 1: Calculate the z-score for Math
For Math:
- Mean change (\(\mu_{Math}\)) = 19,
- Standard deviation (\(\sigma_{Math}\)) = 65,
- Observed score change (\(x\)) = 188.

Using the z-score formula:

\[
z_{Math} = \frac{188 - 19}{65} = \frac{169}{65} \approx 2.6
\]

### Step 2: Calculate the z-score for Verbal
For Verbal:
- Mean change (\(\mu_{Verbal}\)) = 14,
- Standard deviation (\(\sigma_{Verbal}\)) = 50,
- Observed score change (\(x\)) = 188.

Using the z-score formula:

\[
z_{Verbal} = \frac{188 - 14}{50} = \frac{174}{50} \approx 3.48
\]

### Step 3: Interpret the z-scores
- The **z-score for Math** is approximately 2.6.
- The **z-score for Verbal** is approximately 3.48.

The higher the z-score, the further the value is from the mean. Since the z-score for the Verbal score is higher than for the Math score, the **188-point increase is less likely to have occurred in the Verbal test** (because it is more unusual). Therefore, the 188-point increase is more likely to have occurred in the **Math test**.

Would you like further clarification on how z-scores work or their interpretation?

### 5 Relative Questions:
1. What is the significance of a high z-score in data analysis?
2. How is the standard deviation used to measure score variability?
3. Why is it important to normalize data using z-scores in hypothesis testing?
4. Can extreme z-scores (like those close to 3) suggest outliers in data?
5. How would the probabilities change if the standard deviations were smaller?

**Tip:** Z-scores are a useful tool in comparing data points from different distributions, making it easier to assess which results deviate more from their respective means.

Suppose one of the 250 students who paid a private tutor is selected at random. Suppose the selected​ student's score increased on one of the tests by 188 points. Which​ test, the Math or​ Verbal, is the one most likely to have had the 188​-point ​increase?

This problem examines the likelihood of a 188-point score increase in Math or Verbal tests by calculating z-scores. Learn how to interpret z-scores, standard deviations, and apply probability concepts. Suitable for high school students in Grades 11-12.

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