Homework help ### Skittles Color Frequency AnalysisLina suspects that her children are eating her Skittles candy. Skittles come in red, orange, yellow, green, and purple. Her children prefer specific colors, while Lina has no color preference. To determine if her children have been eating her Skittles, she counts the colors in the bowl that she keeps for herself and records the following results:| Color | Frequency ||---------|-----------|| Red | 25 || Orange | 51 || Yellow | 40 || Green | 50 || Purple | 30 |**Key Information:**- $ k = 5 $ (number of possible outcomes)- $ df = 4 $ (degrees of freedom)Skittles colors are produced in equal proportions (20% for each color) before being mixed and bagged. The question is whether the variation in color frequencies is large enough to argue that someone has been eating Lina's Skittles. ### Objective:Test the claim that the underlying color proportions are different from the advertised proportions of 20% each, implying that the color variations in the bowl are not due to random chance. Use a 1% level of significance.#### Step 1: Determine the Hypotheses1. **The outcome for each of the trials (Skittles) above is the color.** - How many possible outcomes ($ k $) are there per trial? - $ k = 5 $2. **What are the hypothetical proportions for each color?** - $ p = \frac{1}{5} $ for each color3. **Construct the null hypothesis using the answer in Question 2.** - This provides the hypothetical proportions of Skittles for each color.4. **What is the alternative hypothesis?** - The alternative hypothesis is not explicitly given in the steps but would generally be: The proportion of Skittles colors does not match the advertised 20% for each color.This set-up helps determine whether Lina's observation of the Skittles colors deviates significantly from what would be expected if no one had been eating specific colors preferentially. ## Step 2: Collect the Data### 5. Observed FrequenciesThe observed frequencies (**O**) are in the table above. Compute the sample size (**n**) by adding the observed frequencies.\[n = \sum O = \_\_\_\_\_\_\_ \]### 6. Expected FrequenciesThe expected frequencies are each equal to the sample size times the respective proportions in the null hypothesis (**E = np**). Fill in the table below with the appropriate expected frequencies.| Color | **O = Observed** | **E = Expected** ||--------|-------------------|------------------|| Red | 25 | \_\_\_\_\_\_ || Orange | 51 | \_\_\_\_\_\_ || Yellow | 40 | \_\_\_\_\_\_ || Green | 50 | \_\_\_\_\_\_ || Purple | 30 | \_\_\_\_\_\_ |### 7. Conditions for Chi-Square DistributionDo the expected frequencies satisfy the conditions of an approximate $\chi^2$ distribution? Explain.---### 8. Degrees of FreedomWhat are the degrees of freedom for this test?\[df = \_\_\_\_\_\_ \]## Step 3: Assess the Evidence### 9. Compute the Chi-Square Test StatisticCompute the $\chi^2$ test statistic,\[\chi^2 = \sum \frac{(O - E)^2}{E},\]by completing the table below.| Color | **O = Observed** | **E = Expected** | $\frac{(O - E)^2}{E}$ ||--------|-------------------|------------------|-------------------------|| Red | 25 | \_\_\_\_\_\_ | \_\_\_\_\_\_ || Orange | 51 | \_\_\_\_\_\_ | \_\_\_\_\_\_ || Yellow | 40 | \_\_\_\_\_\_ | \_\_\_\_\_\_ || Green | 50 | \_\_\_\_\_\_ | \_\_\_\_\_\_ || Purple | 30 | \_\_\_\_\_\_ | \_\

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Description: Homework help ### Skittles Color Frequency AnalysisLina suspects that her children are eating her Skittles candy. Skittles come in red, orange, yellow, green, and purple. Her children prefer specific colors, while Lina has no color preference. To det

It is given that the frequencies of the candies. The claim is that the underlying color proportions…

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Lina has reason to believe that her children are eating her Skittles candy. Skittles come in red, orange, yellow, green, and purple. Her children all have color preferences, but Lina has no color preference. In order to prove the guilt of her children, she counts the colors in the bowl that she keeps out, with the following results.

Lina has reason to believe that her children are eating her Skittles candy. Skittles come in red, orange, yellow, green, and purple. Her children all have color preferences, but Lina has no color preference. In order to prove the guilt of her children, she counts the colors in the bowl that she keeps out, with the following results.

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### Skittles Color Frequency Analysis

Lina suspects that her children are eating her Skittles candy. Skittles come in red, orange, yellow, green, and purple. Her children prefer specific colors, while Lina has no color preference. To determine if her children have been eating her Skittles, she counts the colors in the bowl that she keeps for herself and records the following results:

| Color | Frequency |
|---------|-----------|
| Red | 25 |
| Orange | 51 |
| Yellow | 40 |
| Green | 50 |
| Purple | 30 |

**Key Information:**
- $ k = 5 $ (number of possible outcomes)
- $ df = 4 $ (degrees of freedom)

Skittles colors are produced in equal proportions (20% for each color) before being mixed and bagged. The question is whether the variation in color frequencies is large enough to argue that someone has been eating Lina's Skittles.

### Objective:
Test the claim that the underlying color proportions are different from the advertised proportions of 20% each, implying that the color variations in the bowl are not due to random chance. Use a 1% level of significance.

#### Step 1: Determine the Hypotheses

1. **The outcome for each of the trials (Skittles) above is the color.**
- How many possible outcomes ($ k $) are there per trial?
- $ k = 5 $

2. **What are the hypothetical proportions for each color?**
- $ p = \frac{1}{5} $ for each color

3. **Construct the null hypothesis using the answer in Question 2.**
- This provides the hypothetical proportions of Skittles for each color.

4. **What is the alternative hypothesis?**
- The alternative hypothesis is not explicitly given in the steps but would generally be: The proportion of Skittles colors does not match the advertised 20% for each color.

This set-up helps determine whether Lina's observation of the Skittles colors deviates significantly from what would be expected if no one had been eating specific colors preferentially.

## Step 2: Collect the Data

### 5. Observed Frequencies
The observed frequencies (**O**) are in the table above. Compute the sample size (**n**) by adding the observed frequencies.

\[n = \sum O = \_\_\_\_\_\_\_ \]

### 6. Expected Frequencies
The expected frequencies are each equal to the sample size times the respective proportions in the null hypothesis (**E = np**). Fill in the table below with the appropriate expected frequencies.

| Color | **O = Observed** | **E = Expected** |
|--------|-------------------|------------------|
| Red | 25 | \_\_\_\_\_\_ |
| Orange | 51 | \_\_\_\_\_\_ |
| Yellow | 40 | \_\_\_\_\_\_ |
| Green | 50 | \_\_\_\_\_\_ |
| Purple | 30 | \_\_\_\_\_\_ |

### 7. Conditions for Chi-Square Distribution
Do the expected frequencies satisfy the conditions of an approximate $\chi^2$ distribution? Explain.

---

### 8. Degrees of Freedom
What are the degrees of freedom for this test?

\[df = \_\_\_\_\_\_ \]

## Step 3: Assess the Evidence

### 9. Compute the Chi-Square Test Statistic
Compute the $\chi^2$ test statistic,

\[
\chi^2 = \sum \frac{(O - E)^2}{E},
\]

by completing the table below.

| Color | **O = Observed** | **E = Expected** | $\frac{(O - E)^2}{E}$ |
|--------|-------------------|------------------|-------------------------|
| Red | 25 | \_\_\_\_\_\_ | \_\_\_\_\_\_ |
| Orange | 51 | \_\_\_\_\_\_ | \_\_\_\_\_\_ |
| Yellow | 40 | \_\_\_\_\_\_ | \_\_\_\_\_\_ |
| Green | 50 | \_\_\_\_\_\_ | \_\_\_\_\_\_ |
| Purple | 30 | \_\_\_\_\_\_ | \_\

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