When crossing two types of seeds it is expected to get 4 types of seeds in the ratio of 7:4:3:2 in a sample of 800 seeds 360 were of the first type 220 of the second type 150 of the third type and 70 of the fourth type does this sample fit the proposed distribution? Use the .05 level
To determine if the sample fits the proposed distribution, we need to perform a chi-square test of goodness of fit. This test compares the observed frequencies (the counts of each type of seed in the sample) with the expected frequencies (the counts predicted by the proposed distribution).
Let's go step by step to perform the chi-square test:
Step 1: State the hypotheses:
- Null hypothesis (H0): The sample fits the proposed distribution.
- Alternative hypothesis (Ha): The sample does not fit the proposed distribution.
Step 2: Set the significance level (alpha):
In this case, the significance level is given as 0.05.
Step 3: Calculate the expected frequencies:
Since the proposed distribution ratio is 7:4:3:2, we can calculate the expected frequencies by scaling the ratio to match the total sample size of 800 seeds.
Expected Frequency for the first type = (7/16) * 800
Expected Frequency for the second type = (4/16) * 800
Expected Frequency for the third type = (3/16) * 800
Expected Frequency for the fourth type = (2/16) * 800
Step 4: Calculate the chi-square test statistic (X^2):
The chi-square test statistic can be calculated using the formula: X^2 = Σ((observed - expected)^2 / expected)
Once we have our observed and expected frequencies, we can substitute them in the formula to calculate X^2.
Step 5: Determine the critical value:
To determine the critical value, we need to consult the chi-square distribution table with degrees of freedom equal to the number of categories minus one. In this case, the degrees of freedom is 4 - 1 = 3. We find the critical value at the 0.05 level of significance and 3 degrees of freedom.
Step 6: Compare the calculated chi-square test statistic with the critical value:
If the calculated chi-square test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Now, let's calculate the expected frequencies, the chi-square test statistic, and compare it with the critical value.
Expected Frequency for the first type = (7/16) * 800 = 350
Expected Frequency for the second type = (4/16) * 800 = 200
Expected Frequency for the third type = (3/16) * 800 = 150
Expected Frequency for the fourth type = (2/16) * 800 = 100
Now let's calculate the chi-square test statistic:
X^2 = ((360 - 350)^2 / 350) + ((220 - 200)^2 / 200) + ((150 - 150)^2 / 150) + ((70 - 100)^2 / 100)
After calculating X^2, compare it to the critical value from the chi-square distribution table with 3 degrees of freedom. If X^2 is greater than the critical value, we reject the null hypothesis.
I have outlined the steps to perform the test, and you can follow these steps to calculate the chi-square test statistic and compare it with the critical value to determine if the sample fits the proposed distribution at the 0.05 level of significance.