## Notation

It is crucial to distinguish between sample statistics and population parameters.

| | Sample Statistic | Population Parameter |
|---|---|---|
| Mean | $\bar{x}$ | $\mu$ |
| Standard Deviation | $s$ | $\sigma$ |
| Variance | $s^2$ | $\sigma^2$ |
| Proportion | $p$ | $\pi$ |

## Hypothesis Testing Framework

We assume the null hypothesis is true and look for evidence against it.

- **Null Hypothesis ($H_0$):** Represents "no effect", "no difference", or the status quo. (e.g., "There are no pink cows.")
- **Alternative Hypothesis ($H_A$):** Represents what we claim or suspect is true. (e.g., "There is a pink cow.")

If we find sufficient evidence (data) that is unlikely under $H_0$, we **reject** $H_0$. Otherwise, we **fail to reject** $H_0$. We never "accept" the null.

### Exercise - Open Intro 6.8 revised

Greece has faced a severe economic crisis since the end of 2009. A Gallup poll surveyed 1,000 randomly sampled Greeks in 2011 and found that 25% of them said they would rate their lives poorly enough to be considered "suffering."

### Research Question

A researcher wants to test whether the proportion of Greeks who are suffering is different from 20%. Use a significance level of α = 0.05.

### Part 1: State the Hypotheses

Write the null and alternative hypotheses in terms of the population proportion π.

$$H_0: $$

$$H_a: $$

### Part 2: Calculate the Point Estimate

State the sample statistic and sample size

```{r}

```


### Part 3: Conditions


Check if the conditions required for constructing a confidence interval based on these data are met.


### Part 4: Confidence Interval

Construct a 95% confidence interval for the proportion of Greeks who are "suffering".

### Part 5: Hypothesis Testing Using Test Statistic

Test the hypotheses you wrote in Part 1 using a test statistic (i.e., z statistic).


### Part 6: Hypothesis Testing Using a $P$-Value

Test the hypotheses you wrote in Part 1 using a $p-$value.


### Part 7: Conclusion

Compare the conclusions you made in Parts 4, 5, and 6.

### Part 8: Confidence Level and Confidence Interval

Describe what would happen to the con dence interval if we decided to use a higher confidence level.


### Part 9: Sample Size and Confidence Interval

Describe what would happen to the confidence interval if we used a larger sample.

