Question

Flowers were cultivated by a researcher and yielded the following results: / Blomme word gekweek deur 'n navorser en het die volgende resultate gekry:

120 red flowers with green stamens / 120 rooi blomme met groen meeldrade
48 red flowers with red stamens / 48 rooi blomme met rooi meeldrade
36 yellow flowers with green stamens / 36 geel blomme met groen meeldrade
12 yellow flowers with red stamens / 12 geel blomme met rooi meeldrade

According to a certain theory, the flowers should appear in the proportion 9:3:3:1. Test at a 5% significance level if they did appear in that proportion. / Volgens 'n sekere teorie, moet die blomme verskyn in die verhouding 9:3:3:1. Toets op 'n 5% betekenispeil of hulle wel in daardie verhouding verskyn

Answer 1

To test whether the observed flower counts conform to the expected ratio of 9:3:3:1, we can use the chi-squared goodness-of-fit test.

**Step 1: Define the Observed Counts**
- Red flowers with green stamens: 120
- Red flowers with red stamens: 48
- Yellow flowers with green stamens: 36
- Yellow flowers with red stamens: 12

**Step 2: Calculate Total Counts**
Total observed = 120 + 48 + 36 + 12 = 216

**Step 3: Define Expected Ratios**
The ratio of 9:3:3:1 means:
- 9 parts for red flowers with green stamens
- 3 parts for red flowers with red stamens
- 3 parts for yellow flowers with green stamens
- 1 part for yellow flowers with red stamens

Total ratio parts = 9 + 3 + 3 + 1 = 16

**Step 4: Calculate Expected Counts**
We can calculate the expected counts for each category based on the total count of 216:
- Expected count for red flowers with green stamens = \((9/16) \times 216 = 121.5\)
- Expected count for red flowers with red stamens = \((3/16) \times 216 = 40.5\)
- Expected count for yellow flowers with green stamens = \((3/16) \times 216 = 40.5\)
- Expected count for yellow flowers with red stamens = \((1/16) \times 216 = 13.5\)

**Step 5: Compile Observed and Expected Counts**
- Observed Counts: [120, 48, 36, 12]
- Expected Counts: [121.5, 40.5, 40.5, 13.5]

**Step 6: Compute the Chi-Squared Statistic**
The formula for the chi-squared statistic is:
\[
\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}
\]
Where \(O_i\) is the observed count and \(E_i\) is the expected count.

Calculating each term:
- For 120: \(\frac{(120 - 121.5)^2}{121.5} = \frac{(-1.5)^2}{121.5} = \frac{2.25}{121.5} \approx 0.0185\)
- For 48: \(\frac{(48 - 40.5)^2}{40.5} = \frac{(7.5)^2}{40.5} = \frac{56.25}{40.5} \approx 1.39\)
- For 36: \(\frac{(36 - 40.5)^2}{40.5} = \frac{(-4.5)^2}{40.5} = \frac{20.25}{40.5} \approx 0.500\)
- For 12: \(\frac{(12 - 13.5)^2}{13.5} = \frac{(-1.5)^2}{13.5} = \frac{2.25}{13.5} \approx 0.167\)

Now summing these values:
\[
\chi^2 \approx 0.0185 + 1.39 + 0.500 + 0.167 \approx 2.0755
\]

**Step 7: Determine the Degrees of Freedom (df)**
The degrees of freedom for this test is calculated as:
\(df = k - 1\)
where \(k\) is the number of categories – in this case \(k=4\):
\(df = 4 - 1 = 3\)

**Step 8: Find the Critical Value and Compare**
For a 5% significance level and 3 degrees of freedom, we can look up the critical value from the chi-squared distribution table. The critical value (χ² critical) at \(df=3\) for \(\alpha = 0.05\) is approximately 7.815.

**Step 9: Conclusion**
Since \( \chi^2 \approx 2.0755 \) is less than \(7.815\), we fail to reject the null hypothesis. This means there is not enough evidence to conclude that the flower counts deviate from the expected 9:3:3:1 ratio at the 5% significance level.

Thus, the flowers do appear in the expected proportion.