65 chemical engineering students are from urban/suburban backgrounds.
(Success is defined as a C or better in the course.)
52 of the 65 students succeeded.
Another 55 students were from rural backgrounds; 30 of these students succeeded.
A Is there good evidence that the proportion of students who succeed is different for urban/suburban vs. rural backgrounds?
B) Estimate the difference between the success rates for all urban/suburban and rural students who plan to study chemical engineering. Use 90% confidence.
To determine if there is good evidence that the proportion of students who succeed is different for urban/suburban vs. rural backgrounds, we can perform a hypothesis test. Here's how you can do it:
A) Hypothesis Test:
Step 1: State the null and alternative hypotheses.
The null hypothesis (H0) states that there is no difference in the proportion of students who succeed between urban/suburban and rural backgrounds. The alternative hypothesis (Ha) states that there is a difference in the proportion of students who succeed between the two backgrounds.
H0: Proportion of success for urban/suburban = Proportion of success for rural
Ha: Proportion of success for urban/suburban ≠ Proportion of success for rural
Step 2: Determine the significance level (α).
The significance level is typically set at 0.05 or 5%.
Step 3: Calculate the test statistic.
We can use the Z-test for comparing proportions.
$$Z = \frac{p_1 - p_2}{\sqrt{\frac{{p_1q_1}}{{n_1}} + \frac{{p_2q_2}}{{n_2}}}}$$
where p1 and p2 are the proportions of success, q1 and q2 are (1 - p1) and (1 - p2) respectively, n1 and n2 are the sample sizes for urban/suburban and rural backgrounds.
Step 4: Determine the critical value.
We can use the Z-table or a statistical software to find the critical value for the given significance level.
Step 5: Draw a conclusion.
If the absolute value of the calculated test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Now let's move on to estimating the difference in success rates with 90% confidence.
B) Confidence Interval:
To estimate the difference in success rates between urban/suburban and rural students with 90% confidence, you can use the following formula:
$$\text{{CI}} = (p_1 - p_2) \pm z \times \sqrt{\frac{{p_1q_1}}{{n_1}} + \frac{{p_2q_2}}{{n_2}}}$$
where p1 and p2 are the proportions of success, q1 and q2 are (1 - p1) and (1 - p2) respectively, n1 and n2 are the sample sizes for urban/suburban and rural backgrounds, and z is the critical value corresponding to the desired confidence level (e.g., 1.645 for 90% confidence).
By substituting the appropriate values into the formula, you can calculate the confidence interval. The confidence interval will give you a range of plausible values for the difference in success rates.
Remember, it's important to have accurate and representative data to ensure reliable results in hypothesis testing and confidence interval estimation.