## Lesson 14: Tests of Significance

October 23, 2017

Review:

- Estimation with Confidence Intervals

Presentation:

**Significance Testing**- aka Hypothesis Testing
- Purpose: to evaluate data for evidence of significant agreement or disagreement

**Step 1.**Setup the null hypothesis (Ho) and alternate hypothesis (Ha)**Step 2.**Calculate the appropriate test statistic**Step 3.**Find the P-value (probability of obtaining result by chance)**Step 4.**Interpret results, compare P-value to α = 0.05

**Example A**:**Do Math SAT scores improve significantly with coaching?**- National Math SAT scores are normally distributed with mean score = 505 and std. dev = 62
- Sampled 1,000 students who received coaching
- Sample mean score was 509
- Are these results significantly better than the national average?
**Step 1**: Setup the hypothesis test

**μ = 505, σ = 62**

**x̅ = 509, n = 1,000**

**Ho: μ = 505**

**Ha: μ > 505****Step 2**: Calculate the appropriate test statistic**Z-test =****(x̅ – μ)/(σ/√n)**- (509 – 505)/(62/√1000) = 4/1.96 =
**2.04**

**Step 3**: Find P-value- P = 1 – P(Z<2.04) = 1 – 0.9793 =
**0.0207**

- P = 1 – P(Z<2.04) = 1 – 0.9793 =
**Step 4**: Interpret results- 0.0207 ≅ 2.07% probability of getting this result by chance
- 0.0207 < 0.05
**Reject Ho**- Coaching seems to improve scores significantly

**Example B**:**Has a student paper been plagiarized?**- Previous student papers contain 7 unique vocabulary words on average with std dev of 2.6
- Submitted paper contains 10 unique words
- Is the submitted paper significantly different?
**Step 1**: Setup the hypothesis test**μ = 7, σ = 2.6**

**x̅ = 10**

**Ho: μ = 7**

**Ha: μ > 7**

**Step 2**: Calculate the appropriate test statistic**Z-test =****(x̅ – μ)/(σ/√n)**- (10 – 7)/2.6 =
**1.15**

**Step 3**: Find P-value- P = 1 – P(Z<1.15) = 1 – 0.8749 =
**0.1251**

- P = 1 – P(Z<1.15) = 1 – 0.8749 =
**Step 4**: Interpret results- 0.1251 ≅ 12.51% probability of getting result by chance
- 0.1251 > 0.05
**Fail to Reject Ho**- Unique vocabulary is within normal range, no evidence of plagiarism

**Significance testing is like paternity testing.**- When you check father-child DNA for a match you can prove one person is or is not the father.
- The same test does not prove another person is the father.
- You’re evaluating only one possibility at a time.
- So, when interpreting results, we either “Reject Ho” or “Fail to Reject Ho”

Assignment:

**Problem 1.1.**More than 200,000 people worldwide take the GMAT examination each year as they apply for MBA programs. Their scores vary Normally with mean about**μ = 525**and standard deviation about**σ = 100**. One hundred students,**n = 100**, go through a rigorous training program designed to raise their GMAT scores. The students who go through the program have an average score of**x̅ = 541.4**. Is there evidence to suggest the training program significantly improves GMAT scores?**Problem 1.2.**A newly installed rooftop solar system has been producing energy for**n = 100**days. Average energy production is**41.8**kWh per day with a standard deviation of**13.9**kWh. The solar panel manufacturer claims the panels typically produce**40**kWh per day. Is the newly installed system producing significantly more energy than estimated by the manufacturer?

* Most example and activity problems presented above are derived from* Moore, D.S., McCabe, G.P., and Craig, B.A., 2009. Introduction to the Practice of Statistics, 6th Edition. New York: W.H. Freeman and Company. *