SOLUTION: A poll was taken of 100 college freshmen to determine whether chocolate and vanilla was the preferred ice cream flavor. It was expected that 25% would prefer chocolate and 75% woul
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Question 1177872: A poll was taken of 100 college freshmen to determine whether chocolate and vanilla was the preferred ice cream flavor. It was expected that 25% would prefer chocolate and 75% would prefer vanilla. Lets look at the results of the survey. Instead of 25% of the students polled preferring chocolate, it turns out 39% preferred it,and instead 75% of the students polled preferring vanilla,it turned out that 61% preferred it.
a)Compute x2 (chi-square)
b) state and explain the conditions necessary for the application of X2 (chi-square)
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's solve this problem step-by-step.
**a) Compute the Chi-Square (χ²) Statistic**
1. **Set up the Observed and Expected Frequencies**
* **Total Students:** 100
* **Expected Chocolate:** 25% of 100 = 25
* **Expected Vanilla:** 75% of 100 = 75
* **Observed Chocolate:** 39% of 100 = 39
* **Observed Vanilla:** 61% of 100 = 61
| Flavor | Observed (O) | Expected (E) |
| ----------- | ------------ | ------------ |
| Chocolate | 39 | 25 |
| Vanilla | 61 | 75 |
2. **Calculate the Chi-Square Statistic (χ²)**
* χ² = Σ [(O - E)² / E]
* For Chocolate:
* (39 - 25)² / 25 = (14)² / 25 = 196 / 25 = 7.84
* For Vanilla:
* (61 - 75)² / 75 = (-14)² / 75 = 196 / 75 ≈ 2.6133
* χ² = 7.84 + 2.6133 ≈ 10.4533
* Therefore, χ² ≈ 10.45
**b) Conditions Necessary for the Application of Chi-Square (χ²)**
The chi-square test for goodness of fit or independence relies on several assumptions:
1. **Random Sampling:** The data must be collected through a random sampling method. This ensures that the sample is representative of the population. In our case, it states that a poll of 100 college freshmen was taken.
2. **Independence:** The observations must be independent of each other. This means that one observation does not influence another. In our case, the ice cream preference of one student should not influence the preference of another student.
3. **Categorical Data:** The data must be categorical. In our case, the ice cream preferences are categorized as chocolate or vanilla.
4. **Expected Frequencies:** All expected frequencies must be greater than or equal to 5. This is a crucial condition. If any expected frequency is less than 5, the chi-square approximation becomes unreliable. In our case, both expected frequencies (25 and 75) are greater than 5.
5. **Sample Size:** The sample size must be large enough. There is no specific rule, but a larger sample size generally leads to a more accurate chi-square approximation.
6. **Mutually Exclusive Categories:** The categories must be mutually exclusive. In our case, a student can only prefer one of the two flavors.
**Explanation of the Conditions**
* **Random Sampling:** Ensures the sample accurately represents the population, reducing bias.
* **Independence:** Prevents skewed results due to related data points.
* **Categorical Data:** Chi-square is designed for categorical variables, not continuous ones.
* **Expected Frequencies:** Small expected frequencies can lead to inflated chi-square values, making it more likely to reject the null hypothesis when it is true.
* **Sample Size:** A larger sample size improves the reliability of the chi-square test.
* **Mutually Exclusive Categories:** Avoids overlapping or ambiguous categories.
In our ice cream example, all of these conditions are met, so the chi-square test is appropriate.
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