I need help finding the null and alternative hypothesis with this project I am doing. I am surveying 200 students about their academic performance. 45 students took an online class and passed, 55 took online and didn't pass. 85 took in-person and passed, while 15 took in-person and didn't pass. I also need help finding the test statistic and p-value as well.



To address your project, let's break down the hypothesis testing process. Here are the steps we will follow:

1. **State the null and alternative hypotheses.**
2. **Calculate the test statistic.**
3. **Find the p-value.**

### 1. State the Hypotheses

We are interested in comparing the success rates of students who took online classes versus those who took in-person classes.

- **Null Hypothesis (H₀)**: The proportion of students who pass is the same for online and in-person classes. \( H₀: p_{\text{online}} = p_{\text{in-person}} \)
- **Alternative Hypothesis (H₁)**: The proportion of students who pass is different for online and in-person classes. \( H₁: p_{\text{online}} \neq p_{\text{in-person}} \)

### 2. Calculate the Test Statistic

First, let's summarize the dataTo address your project, let's break down the hypothesis testing process. Here are the steps we will follow:

1. **State the null and alternative hypotheses.**
2. **Calculate the test statistic.**
3. **Find the p-value.**

### 1. State the Hypotheses

We are interested in comparing the success rates of students who took online classes versus those who took in-person classes.

- **Null Hypothesis (H₀)**: The proportion of students who pass is the same for online and in-person classes. \( H₀: p_{\text{online}} = p_{\text{in-person}} \)
- **Alternative Hypothesis (H₁)**: The proportion of students who pass is different for online and in-person classes. \( H₁: p_{\text{online}} \neq p_{\text{in-person}} \)

### 2. Calculate the Test Statistic

First, let's summarize the 

- Online:
  - Passed: 45
  - Didn't pass: 55
  - Total: 100
- In-person:
  - Passed: 85
  - Didn't pass: 15
  - Total: 100

The test statistic for comparing two proportions can be calculated using the formula for a two-proportion z-test:

\[ 
\hat{p}_{\text{online}} = \frac{45}{100} = 0.45 
\]
\[ 
\hat{p}_{\text{in-person}} = \frac{85}{100} = 0.85 
\]
\[ 
\hat{p} = \frac{45 + 85}{200} = \frac{130}{200} = 0.65 
\]

The test statistic (z) is calculated as:

\[
z = \frac{(\hat{p}_{\text{online}} - \hat{p}_{\text{in-person}})}{\sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_{\text{online}}} + \frac{1}{n_{\text{in-person}}} \right)}}
\]

Substituting the values:

\[
z = \frac{(0.45 - 0.85)}{\sqrt{0.65 \times 0.35 \left( \frac{1}{100} + \frac{1}{100} \right)}}
\]
\[
z = \frac{-0.40}{\sqrt{0.65 \times 0.35 \times 0.02}}
\]
\[
z = \frac{-0.40}{\sqrt{0.00455}}
\]
\[
z = \frac{-0.40}{0.0675}
\]
\[
z \approx -5.93
\]

### 3. Find the p-value

To find the p-value, we need to look at the standard normal distribution (z-distribution) for the calculated z-value.

For \( z \approx -5.93 \), the p-value is extremely small (less than 0.0001). 

### Conclusion

Since the p-value is very small, we reject the null hypothesis. This indicates that there is a significant difference between the passing rates of students taking online classes and those taking in-person classes.

Would you like more details or have any questions about this process?

Here are 5 questions you might ask next:

1. How do I interpret the p-value in this context?
2. What assumptions are we making for this hypothesis test?
3. How would the results change if the sample sizes were different?
4. Can we use a different statistical test for this data?
5. What are some potential limitations of this analysis?

**Tip:** When comparing proportions, ensure your sample sizes are large enough for the normal approximation to be valid. If not, consider using a different test, such as Fisher's exact test.

Explore hypothesis testing by comparing the success rates of students in online and in-person classes. Calculate test statistics, p-values, and interpret the results based on survey data of 200 students. Learn the two-proportion z-test formula and its application. Suitable for advanced undergraduate students and researchers.

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