Research at the University of Toledo indicates that 50 percent of the students change their major area of study after their first year in a program. A random sample of 100 students in the College of Business revealed that 48 had changed their major area of study after their first year of the program. Has there been a significant decrease in the proportion of students who change their major after the first year in this program? Test at the .05 level of significance.
To determine if there has been a significant decrease in the proportion of students who change their major after the first year in the College of Business program, we can conduct a hypothesis test.
First, let's define our null and alternative hypotheses:
Null hypothesis (H0): There has been no significant decrease in the proportion of students who change their major after the first year.
Alternative hypothesis (Ha): There has been a significant decrease in the proportion of students who change their major after the first year.
In this case, the null hypothesis suggests that the proportion of students who change their major after the first year is still 50%, while the alternative hypothesis suggests that the proportion is less than 50%.
Next, we need to calculate the test statistic and the critical value.
The test statistic used for this hypothesis test is the z-test, which compares the sample proportion (48/100 = 0.48) to the expected proportion (0.50). The formula for the test statistic is:
z = (p - P) / sqrt(P(1-P) / n)
where p is the sample proportion, P is the expected proportion, and n is the sample size.
In this case, p = 0.48, P = 0.50, and n = 100. Plugging these values into the formula, we can calculate the test statistic.
z = (0.48 - 0.50) / sqrt(0.50(1-0.50) / 100)
= -0.02 / sqrt(0.50*0.50 / 100)
= -0.02 / sqrt(0.25 / 100)
= -0.02 / 0.05
= -0.4
To find the critical value, we can refer to the z-table and look up the z-value for a significance level of 0.05 (assuming a one-tailed test since we are testing for a decrease). The critical value for a significance level of 0.05 is -1.645.
Since the test statistic (-0.4) is greater than the critical value (-1.645), we fail to reject the null hypothesis. This means that we do not have enough evidence to conclude that there has been a significant decrease in the proportion of students who change their major after the first year in the College of Business program at a significance level of 0.05.
In simpler terms, we cannot say with 95% confidence that there has been a significant decrease in the proportion of students changing their major after the first year in the College of Business program based on the given sample of 100 students.