Problem 26 - Basic Statistics for Business and Economics with Student

Question

Research at the University of Toledo indicates that 50 percent of 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.

Step-by-Step Solution

Answer

No significant decrease; fail to reject null hypothesis.

Step 1: Determine Null and Alternative Hypotheses

First, we identify our null and alternative hypotheses. The null hypothesis ( H_0) is that the proportion of students who change their major is 50%, or p = 0.5. The alternative hypothesis ( H_a) is that the proportion is less than 50%, or p < 0.5. This is because we're looking for a decrease in the proportion.

Step 2: Determine the Significance Level and Find Critical Value

We are given a significance level α = 0.05. For a one-tailed test using the standard normal distribution, we need to find the critical value corresponding to this significance level. In this case, the critical z-value is approximately -1.645.

Step 3: Calculate the Test Statistic

Use the formula for the z-test for proportions:\[ z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} \]Where \(\hat{p}\) is the sample proportion (48/100 = 0.48), \(p\) is the population proportion (0.5), and \(n\) is the sample size (100).Substitute and solve:\[z = \frac{0.48 - 0.5}{\sqrt{\frac{0.5 \times 0.5}{100}}} = \frac{-0.02}{0.05} = -0.4\]

Step 4: Compare Test Statistic to Critical Value

Compare the calculated z-test statistic (-0.4) to the critical z-value (-1.645). Since -0.4 is not less than -1.645, we fail to reject the null hypothesis.

Step 5: State the Conclusion

Since we did not reject the null hypothesis, there is no significant evidence at the 0.05 level to suggest that the proportion of students changing their major after the first year has decreased from 50% in this sample of College of Business students.

Key Concepts

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement that suggests there is no effect or no difference in the population. It acts as the default assumption until evidence suggests otherwise. In our exercise, the null hypothesis ( H_0) posits that the proportion of students changing their major is 50%, which means that half of the students tend to switch their field of study after the first year.
This hypothesis is essential because it provides a baseline comparison for evaluating the sample data. It helps us determine if any observed effect in the data, such as a change in the proportion of students switching majors, is statistically significant or if it can be attributed to random chance.

Alternative Hypothesis

The alternative hypothesis ( H_a) is what researchers typically aim to prove. It states that there is an effect or a difference. In our case, the alternative hypothesis suggests that less than 50% of students are changing their major after their first year.
This hypothesis is concerned with detecting whether there is a decrease in the number of students changing their majors compared to the stated 50%. It provides a direct contrast to the null hypothesis. If evidence supports the alternative hypothesis, it indicates that the sample provides enough proof of a significant change in student behavior.

Z-test for Proportions

The z-test for proportions is a statistical test used to determine if the proportion of a sample differs significantly from the assumed population proportion. It's particularly useful when working with categorical data.

• Sample Proportion (\(\hat{p}\)): The proportion observed in the sample.
• Population Proportion (\(p\)): The assumed proportion in the population as stated by the null hypothesis.
• Formula: \[ z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} \]

In this exercise, \(\hat{p} = 0.48\) and \(p = 0.5\). By calculating the z-value from these proportions, we can compare it to a critical z-value to decide whether to support or reject our null hypothesis. The z-test assesses whether the observed data can occur given the null hypothesis assumption or if it's an unlikely event when the null is true.

Significance Level

The significance level, often denoted as alpha (\(\alpha\)), is the criterion used for deciding whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs if the null hypothesis is incorrectly rejected when it is actually true. Commonly used significance levels are 0.05, 0.01, and 0.10.
In this problem, we use a significance level of 0.05, implying a 5% risk of mistakenly rejecting the null hypothesis. At this level, any z-value below the critical z-value (here, -1.645) will lead us to reject the null hypothesis. But since our calculated z-value is -0.4, which is not in the rejection region, we maintain the null hypothesis and conclude there's no significant decrease in the proportion of students switching their majors.