Final answer:
To determine if the mean tuition in private colleges is more than PHP 2,500 higher than in state colleges, a hypothesis test using the provided sample data is performed. The test will include calculating the test statistic and comparing it to the critical value at a 5% significance level to make a decision about the null hypothesis. Type I and Type II errors are also considered in the interpretation of the results.
Step-by-step explanation:
The student is asking to determine if the mean tuition per semester in the private colleges and universities is more than PHP 2,500 higher compared to state colleges and universities in the National Capital Region. To do this, a hypothesis test for the difference in means can be conducted using the given sample data: a sample mean tuition of PHP 25,000 for private colleges with a standard deviation of PHP 4,500 from a sample of 20 institutions, and a sample mean tuition of PHP 15,900 for state colleges with a standard deviation of PHP 3,900 from a sample of 15 institutions. A 5% significance level is used for the test.
First, we calculate the standard error for the difference in sample means, then determine the test statistic using a formula that accounts for the unequal sample sizes and standard deviations. Next, using the degrees of freedom calculated based on the given standard deviations and sample sizes, we consult the t-distribution table to find the critical value at a 5% significance level. If our test statistic exceeds the critical value, we reject the null hypothesis and conclude that there is a statistically significant difference in tuition costs more than PHP 2,500.
Understanding the concepts of Type I and Type II errors is also crucial. A Type I error would occur if we incorrectly reject the null hypothesis, concluding that the tuition difference is indeed greater than PHP 2,500 when it is not. Conversely, a Type II error would occur if we fail to reject the null hypothesis when the tuition difference is in fact greater than PHP 2,500.