Key to Student Loan Case PDF

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An alternate assignment that helps with case studies (Dr. Miller)...

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MTH 220 – Introduction to Statistics Practice Case Like Case 1B Student Loans Case Study: A recent study was focused on students at state-funded higher education institutions in Texas. Past studies of such students had concluded that one-third of students attending public higher educational facilities in Texas had loans in excess of $20K in order to pay for their education. The current study is investigating whether or not this percentage has risen. To investigate the possibility of this, a survey of 1280 student loan borrowers found that 478 had loans totaling more than $20K. Is this sample proportion sufficient statistical evidence to conclude that more than one-third of students in the population owe over $20K from the costs of college?

MTH 220 – Introduction to Probability & Statistics Recap of Key Ingredients for the Statistical Investigation of Case 1B

Step in the Process

Student Loan Case Specifics

Define the Population Under Study All students that attend a state-funded higher education institution in Texas Obtain a Sample – Investigate if it is Representative of Population

The 1280 students who attend a public college or university in Texas that were part of the survey. There is nothing written in the problem statement that makes us believe the sample isn’t representative.

State the Variable of Interest The variable of interest is Bernoulli. Success means that the student has over $20K in student loans (the interest in the problem). Failure means that the student does not have over $20K in student loans. Success is assigned the number “1” and failure is assigned the outcome “0”. Each . variable can be called X i and i  1 Define a Parameter to Investigate p = the proportion of students who attend a state-funded college or university in Texas that have over $20K in student loans. Estimate the Parameter w/ a Statistic pˆ = the proportion of students in the sample of 1280 which responded to the survey stating that they had over $20K in student loans in order to pay for their education. From the survey results, we can calculate that pˆ  478 1280

State the Hypotheses to be Tested

H0 : p  1 3 (the fraction of students in the population with over $20K in loans hasn’t changed since the last study) H A : p 1 3 (the fraction of students in the population with over $20K in loans is higher than seen in the past)

Determine a Test Statistic to Use for the Hypothesis Test Obtain the Sampling Distribution of the Test Statistic

The test statistic is the total number of successes observed in the sample,  X i The EXACT sampling distribution is binomial with n  1280 and p  1 3 . This is a binomial experiment since we have 1280 independent Bernoulli trials (one person needing to take out loans doesn’t affect the chance that others will or won’t), we have assumed p  1 3 and we are interested in the total number of successes. (It is 478 in our sample). So, the requirements for a binomial experiment are satisfied. However, the number of trials is large and so the binomial sampling distribution can be approximated by a normal curve with mean   np  426.67 and

  np 1  p  16.87 Compute the p-value of the Test The p-value is the chance of observing our test statistic value or one more extreme if the null hypothesis is true. We observed “478” successes. Extreme is interpreted as the direction of H A . So, we need the chance of 478 or more successes since the direction of the alternative is “>”. Let X denote number of total successes. Then, we desire P  X  478 . To approximate this p-value, we use standardization:  X  426.67 478  426.67  P  X  478   P    16.87  16.87   P Z  3.04 We know that it is rare to see an outcome of a normal random variable more than 2-3 standard deviations away from the mean. Our normal table, or Excel, or calculator can be used to obtain  P  Z  3.04  1 .9988  .0012

Reach a Conclusion Based on the Pvalue. State it in context of the problem.

The p-value is very small, so we should side with “change”. That is, we have sufficient evidence to reject the null hypothesis and conclude that the alternative hypothesis is most reasonable. This means that the survey contains sufficient statistical evidence to conclude that more than one-third of students in the population will require over $20K in loans as they complete their education....