You are interested in seeing if a new test will help to determine who will be successful in a training program. The current rate of success is 70% for 200 students. When the new test is implemented, the rate increases to 85% for the 50 students who pass the test and are allowed into the program. Is this difference significant?
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To determine if the difference in success rates is significant, we need to conduct a hypothesis test.
First, let's define the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis assumes that there is no significant difference in success rates before and after the new test is implemented. The alternative hypothesis proposes that there is a significant difference in success rates after implementing the new test.
H0: The success rate before implementing the new test is equal to the success rate after implementing the new test.
H1: The success rate before implementing the new test is not equal to the success rate after implementing the new test.
Next, we can calculate the expected success rates under the null hypothesis. Since the overall success rate is currently 70% for 200 students, the expected number of successful students can be calculated as 0.70 * 200 = 140. Furthermore, since 50 students pass the new test and are allowed into the program, we expect the success rate after implementing the new test to be 50/50 = 1.00, or 100%.
Now we can use the chi-square test statistic to compare the observed data (85% success rate) with the expected data (100% success rate under the null hypothesis). The chi-square test will indicate how likely the observed difference is due to chance alone.
The formula to calculate the chi-square test statistic is:
χ^2 = Σ ((O - E)^2 / E)
where:
χ^2 is the chi-square test statistic
O is the observed frequency
E is the expected frequency
In this case, we have one observed frequency of 85% and one expected frequency of 100%.
χ^2 = ((85 - 100)^2 / 100) = 225 / 100 = 2.25
To determine if the difference is significant, we need to compare the calculated chi-square value to a critical value from the chi-square distribution table. The critical value depends on the significance level (usually denoted as α) chosen for the test. Commonly used significance levels are 0.05 and 0.01.
If the calculated chi-square value is greater than the critical value at the chosen significance level, we reject the null hypothesis and conclude that there is a significant difference in success rates before and after implementing the new test.
So, in order to make a conclusive determination on whether this difference is significant, you will need to compare the calculated chi-square value of 2.25 to the critical value at the chosen significance level (usually 0.05 or 0.01).