A researcher conducts a test on the effectiveness of a cholesterol treatment on 114 total subjects. Assuming the tails of distributions are normal distribution, is there evidence that the treatment is effective?
Cholesterol Decreased
No Cholesterol Decrease
Total
Treatment
38
18
56
No treatment
30
28
58
Total
68
46
114
To determine if there is evidence that the cholesterol treatment is effective, we can use a hypothesis test. Specifically, we can conduct a chi-squared test for independence.
Step 1: Formulate the null and alternative hypotheses:
- Null hypothesis (H0): The cholesterol treatment has no effect and is not related to cholesterol decrease.
- Alternative hypothesis (Ha): The cholesterol treatment is effective and is related to cholesterol decrease.
Step 2: Set the significance level:
The significance level, denoted as α, determines how much evidence is required to reject the null hypothesis. Let's assume a significance level of α = 0.05, which is common in statistical analysis.
Step 3: Calculate the expected frequencies:
Under the assumption of independence between the cholesterol treatment and cholesterol decrease, we can calculate the expected frequencies for each cell of the contingency table. The expected frequency for each cell is calculated by multiplying the row total by the column total and dividing by the grand total.
For example, to calculate the expected frequency for the top left cell, we would do the following:
Expected Frequency = (row total / grand total) * (column total / grand total) * grand total
Step 4: Calculate the chi-squared test statistic:
The chi-squared test statistic measures how much the observed frequencies differ from the expected frequencies, assuming the null hypothesis is true. The formula for the chi-squared test statistic is:
χ² = Σ ((Observed Frequency - Expected Frequency)^2) / Expected Frequency
For each cell in the contingency table, calculate the (Observed Frequency - Expected Frequency)^2 / Expected Frequency and sum them all up.
Step 5: Determine the critical value or p-value:
To determine whether the observed chi-squared test statistic is statistically significant, we compare it to the critical value from the chi-squared distribution with degrees of freedom equal to (rows - 1) * (columns - 1). Alternatively, we can calculate the p-value associated with the chi-squared test statistic.
Step 6: Make a decision:
If the calculated chi-squared statistic is greater than the critical value or the p-value is less than the significance level (α), we reject the null hypothesis in favor of the alternative hypothesis. This indicates that there is evidence to suggest that the cholesterol treatment is effective. Otherwise, if the chi-squared statistic is less than the critical value or the p-value is greater than α, we fail to reject the null hypothesis, suggesting there is not enough evidence to support the effectiveness of the cholesterol treatment.
By following these steps and applying the chi-squared test for independence to the given data, you can determine if there is evidence that the cholesterol treatment is effective.