https://file.aiquickdraw.com/mathbot/file/173030484124487zialzr.jpg

I'll analyze the problems given in the photo:

### Problem Breakdown

1. **Problem 1 (10 points)**  
   Test the claim that less than \( \frac{1}{3} \) of all **peanuts** M&M's are orange.
   
   - This requires setting up a hypothesis test for a population proportion, where the null hypothesis \( H_0 \) could state that the proportion of orange M&Ms in the peanuts type is \( \frac{1}{3} \) or higher, while the alternative hypothesis \( H_1 \) will state it is less than \( \frac{1}{3} \).

2. **Problem 2 (10 points)**  
   Test the claim that the probability of getting a red **classic** M&M is \( \frac{1}{8} \).
   
   - Here, the null hypothesis \( H_0 \) would state that the probability of drawing a red classic M&M is exactly \( \frac{1}{8} \). The alternative hypothesis \( H_1 \) would state that this probability is not equal to \( \frac{1}{8} \).

3. **Problem A (20 points)**  
   Perform a Chi-squared Goodness of Fit test to determine if the actual color distribution of **classic** M&M's aligns with the given distribution:
   
   | Color   | Expected Proportion |
   |---------|----------------------|
   | Orange  | 0.4                 |
   | Yellow  | 0.15                |
   | Red     | 0.10                |
   | Green   | 0.15                |
   | Blue    | 0.10                |
   | Brown   | 0.10                |
   
   - This will require comparing observed and expected frequencies for each color category to see if the actual distribution matches the expected proportions.

4. **Problem B (20 points)**  
   Perform a Chi-squared Test of Independence to determine if the color of M&M's is independent of the type (classic or peanuts).
   
   - This is a test to see if there is an association between M&M color and type (classic or peanuts). We'll compare observed frequencies in each color/type combination with the expected frequencies assuming independence.

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### Relative Questions:
1. How is a hypothesis test for a proportion different from a test for independence?
2. What are the steps involved in calculating a Chi-squared Goodness of Fit test?
3. Why is the Chi-squared distribution used in tests for independence and goodness of fit?
4. How do we calculate the expected frequencies in a Chi-squared Test of Independence?
5. How would the conclusions change if we used a significance level other than \( \alpha = 0.05 \)?

### Tip:
When performing Chi-squared tests, ensure the expected frequency for each category is at least 5 to meet the assumptions of the test for accuracy.

Test several statistical hypotheses and perform Chi-squared tests based on M&M color distribution and type data.

Explore hypothesis testing and Chi-squared tests with a dataset on M&M colors and types. This solution covers goodness of fit and test of independence, with specific hypothesis testing for proportions in M&M distribution. Ideal for undergraduate statistics students.

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