5) In a study of 48 subjects who used Huntington Tutoring Service to increase their statistics scores, each subject spent $400 and after 9 weeks, the mean score went up from 64 to 65 points (out of 100).

There is a 2.99% likelihood that such results are due to chance. Huntington Tutoring Service guaranteed that after 12 weeks, students are expected to increase their overall score by at least 33 points.

a) Do the results appear to have practical significance?

Explain.

Without the correct explanation, your answer earns zero credit.

b) Do the results appear to have statistical significance? Explain.

Without the correct explanation, your answer earns zero credit.

### Part (a): **Practical Significance**

**Practical significance** refers to whether the effect of a result is large enough to be meaningful in a real-world context. In this case, the **mean score increased by only 1 point** (from 64 to 65) after 9 weeks, despite each student spending $400 on tutoring.

- While the score did increase slightly, a **1-point improvement** is minimal compared to the expected improvement guaranteed by Huntington Tutoring Service, which promised an increase of **33 points after 12 weeks**.
- The cost of $400 for such a small improvement does not seem to justify the investment, as it would not be considered a meaningful or **practically significant** result for students seeking substantial gains in their scores.

**Conclusion**: 
The results do **not appear to have practical significance**, as the increase in score (1 point) is too small to be meaningful given the cost of the tutoring service and the initial expectations of improvement.

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### Part (b): **Statistical Significance**

**Statistical significance** relates to the likelihood that the observed result (an increase of 1 point) occurred by chance. A **2.99% likelihood** (or a **p-value of 0.0299**) means that there is only a 2.99% chance that the result was due to random variation, which is below the conventional threshold of **5% (p < 0.05)** used to determine statistical significance.

- Since the p-value is less than 0.05, this means that the observed increase in score is **unlikely to have occurred by chance**.
  
**Conclusion**: 
The results do have **statistical significance** because the p-value of 0.0299 indicates that the increase in scores is unlikely to be due to random chance, even though the effect is very small.

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Would you like further details or clarification on any part of this explanation?

Here are five questions to extend your understanding:
1. What is the difference between statistical and practical significance?
2. How is a p-value interpreted in hypothesis testing?
3. What factors contribute to practical significance?
4. Why might a result be statistically significant but not practically significant?
5. How could the tutoring service improve the practical significance of their results?

**Tip**: Statistical significance tells us if a result is unlikely due to chance, but practical significance evaluates if the result is meaningful in real-world terms.

In a study of 48 subjects who used Huntington Tutoring Service to increase their statistics scores, each subject spent $400 and after 9 weeks, the mean score went up from 64 to 65 points (out of 100). There is a 2.99% likelihood that such results are due to chance. Huntington Tutoring Service guaranteed that after 12 weeks, students are expected to increase their overall score by at least 33 points.
a) Do the results appear to have practical significance? Explain.
b) Do the results appear to have statistical significance? Explain.

Explore the statistical and practical significance of a study involving 48 subjects using Huntington Tutoring Service. Learn about statistical significance with a 2.99% chance the results are due to chance (p-value 0.0299), and why the 1-point increase may lack practical significance despite the investment. Suitable for undergraduate statistics students.

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