Question

the formula: \\(\sum \frac{(o-e)^2}{e}\\) \\(\sigma\\) = sum of \\(o\\) = observed values \\(e\\) = expected values null hypothesis (\\(h_0\\)): my experimental condition causes no statistical difference in my observed versus expected data alternative hypothesis (\\(h_a\\)): my experimental condition causes a statistical difference in my observed versus expected data p-value and degrees of freedom degrees of freedom: # of variables - 1 (variables include color, genotype, stage, etc) our goal is to be 95% sure that we are either rejecting or failing to reject our null, so we look at the 0.05 column. when you use the p-value of 0.05, you are saying that you would only get results this extreme 5% of the time in your experiment. (aka: there is only a 5% chance that your results are due to chance) if our calculated number is: - higher than our critical value in the 0.05 column: reject the null hypothesis - lower than our critical value in the 0.05 column: fail to reject the null hypothesis chart: degrees of freedom vs probability table with rows (1–10) and columns (0.95, 0.90, ..., 0.001) script to follow: (fail to reject / reject) my null hypothesis because the calculated chi square value of ____ is (lower than / higher than) the critical value of __ for p=0.05 at ____ degrees of freedom. yours does not have to be this exactly, but this does cover everything you need to include **

Explanation:

To solve a chi - square test problem, we follow these steps:

Step 1: Identify the number of variables

First, we need to determine the number of variables in the experiment. Let's assume we have \(n\) variables. Then, the degrees of freedom (\(df\)) is calculated using the formula \(df=\text{number of variables}- 1\).

Step 2: Calculate the chi - square value

We use the formula \(\chi^{2}=\sum\frac{(O - E)^{2}}{E}\), where \(O\) is the observed frequency and \(E\) is the expected frequency. We calculate \((O - E)\) for each category, square it, divide by \(E\), and then sum up all these values to get the chi - square statistic.

Step 3: Find the critical value

Using the degrees of freedom (\(df\)) calculated in Step 1, we look at the chi - square distribution table (the one provided in the problem) and find the critical value corresponding to a significance level of \(\alpha = 0.05\).

Step 4: Make a decision

We compare the calculated chi - square value (\(\chi^{2}_{\text{calculated}}\)) with the critical value (\(\chi^{2}_{\text{critical}}\)):

• If \(\chi^{2}_{\text{calculated}}>\chi^{2}_{\text{critical}}\) (for \(\alpha = 0.05\)), we reject the null hypothesis. This means that there is a statistically significant difference between the observed and expected data, and our experimental condition has an effect.
• If \(\chi^{2}_{\text{calculated}}<\chi^{2}_{\text{critical}}\) (for \(\alpha = 0.05\)), we fail to reject the null hypothesis. This implies that there is no statistically significant difference between the observed and expected data, and our experimental condition does not seem to have a significant effect.

For example, if we have 3 variables, then \(df=3 - 1=2\). The critical value for \(df = 2\) and \(\alpha=0.05\) from the table is \(5.99\). If our calculated chi - square value is \(6.5\) (which is greater than \(5.99\)), we would say: "Reject my null hypothesis because the calculated chi square value of \(6.5\) is higher than the critical value of \(5.99\) for \(p = 0.05\) at \(2\) degrees of freedom." If the calculated value was \(5.0\) (less than \(5.99\)), we would say: "Fail to reject my null hypothesis because the calculated chi square value of \(5.0\) is lower than the critical value of \(5.99\) for \(p = 0.05\) at \(2\) degrees of freedom."

Since the problem does not provide specific observed (\(O\)) and expected (\(E\)) values, as well as the number of variables, we can't give a numerical answer for the chi - square test result. But the above steps outline the general process to solve a chi - square hypothesis testing problem.

Answer:

To solve a chi - square test problem, we follow these steps:

Step 1: Identify the number of variables

First, we need to determine the number of variables in the experiment. Let's assume we have \(n\) variables. Then, the degrees of freedom (\(df\)) is calculated using the formula \(df=\text{number of variables}- 1\).

Step 2: Calculate the chi - square value

We use the formula \(\chi^{2}=\sum\frac{(O - E)^{2}}{E}\), where \(O\) is the observed frequency and \(E\) is the expected frequency. We calculate \((O - E)\) for each category, square it, divide by \(E\), and then sum up all these values to get the chi - square statistic.

Step 3: Find the critical value

Using the degrees of freedom (\(df\)) calculated in Step 1, we look at the chi - square distribution table (the one provided in the problem) and find the critical value corresponding to a significance level of \(\alpha = 0.05\).

Step 4: Make a decision

We compare the calculated chi - square value (\(\chi^{2}_{\text{calculated}}\)) with the critical value (\(\chi^{2}_{\text{critical}}\)):

• If \(\chi^{2}_{\text{calculated}}>\chi^{2}_{\text{critical}}\) (for \(\alpha = 0.05\)), we reject the null hypothesis. This means that there is a statistically significant difference between the observed and expected data, and our experimental condition has an effect.
• If \(\chi^{2}_{\text{calculated}}<\chi^{2}_{\text{critical}}\) (for \(\alpha = 0.05\)), we fail to reject the null hypothesis. This implies that there is no statistically significant difference between the observed and expected data, and our experimental condition does not seem to have a significant effect.

For example, if we have 3 variables, then \(df=3 - 1=2\). The critical value for \(df = 2\) and \(\alpha=0.05\) from the table is \(5.99\). If our calculated chi - square value is \(6.5\) (which is greater than \(5.99\)), we would say: "Reject my null hypothesis because the calculated chi square value of \(6.5\) is higher than the critical value of \(5.99\) for \(p = 0.05\) at \(2\) degrees of freedom." If the calculated value was \(5.0\) (less than \(5.99\)), we would say: "Fail to reject my null hypothesis because the calculated chi square value of \(5.0\) is lower than the critical value of \(5.99\) for \(p = 0.05\) at \(2\) degrees of freedom."

Since the problem does not provide specific observed (\(O\)) and expected (\(E\)) values, as well as the number of variables, we can't give a numerical answer for the chi - square test result. But the above steps outline the general process to solve a chi - square hypothesis testing problem.