Summary of the Lecture on Hypothesis Testing
In today's class, we discussed the process of hypothesis testing using a real-world example. Faye, motivated by findings on bilingualism in the broader American population, decided to test if the proportion of bilinguals in her city was greater than the national average of 26%. We went through the steps of setting up and testing hypotheses, calculating the sample proportion, the test statistic, and finally the p-value.
Key Points from the Lecture
Setting Up the Hypotheses
- Null Hypothesis (H0): The proportion of bilinguals in her city is equal to the national average, ( p = 0.26 ).
- Alternative Hypothesis (HA): The proportion of bilinguals in her city is greater than the national average, ( p > 0.26 ).
Data Collection and Sample Proportion
- Faye took a sample of 120 people from her city.
- 40 out of these 120 people could speak more than one language.
- Sample Proportion ( \hat{p} ): ( \frac{40}{120} = \frac{1}{3} \approx 0.33 ).
Calculation of Test Statistic
- Test Statistic (Z): Calculated to be approximately 1.83.
- Formula used was:
[
Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}
]
where ( \hat{p} ) is sample proportion, ( p_0 ) is assumed population proportion (0.26), and ( n ) is sample size (120).
P-Value Calculation
- P-Value: Represents the probability of observing a test statistic at least as extreme as the one observed if the null hypothesis is true.
- Since the test was one-tailed (we only care if the sample proportion is greater), the p-value calculated was approximately 0.0336 (or 3.36%).
Decision Making
- Faye needs to compare the p-value with a predetermined significance level (alpha, ( \alpha )).
- If ( \alpha = 0.05 ) (5%), since 0.0336 < 0.05, Faye would reject the null hypothesis, suggesting that the proportion of bilinguals in her city is significantly greater than the national average.
- If ( \alpha = 0.01 ) (1%), since 0.0336 > 0.01, Faye would fail to reject the null hypothesis.
Considerations and Assumptions
- The calculations assume that the sample is randomly selected and the sample size is large enough to approximate the sampling distribution of the proportion as normal (Central Limit Theorem).
- Necessary conditions for these tests include randomness, normality, and independence among sampled data points.
This analysis concluded that according to her sample and given a typical alpha level of 5%, Faye’s hypothesis that her city has a higher proportion of bilinguals than the national average could be supported.