A parent advisory board for a certain university was concerned about the effect of part-time jobs on the academic achievement of students attending the university. To obtain some information, the advisory board surveyed a simple random sample of 200 of the more than 20,000 students attending the university. Each student reported the average number of hours spent working part-time each week and his or her perception of the effect of part-time work on academic achievement. The data in the table below summarize the students’ responses by average number of hours worked per week (less than 1 11 to 20, more than 20) and perception of the effect of part-time work on academic achievement (positive, no effect, negative) Average Time Spent on Part-Time Jobs Than 11 11 to 20 Hours More Than 20 Hours per Week Hours per Week per Week 21 Perception of the Effect of Part- Time Work on Academic No Effect Positive Effect tive Effect 18 A chi-square test was used to determine if there is an association between the effect of part-time work on academic achievement and the average number of hours per week that students work. Computer output that resulted from performing this test is shown below CHI-SQUARE TEST Expected counts are printed below observed counts 11 11-20 20 Total 35 16975 11200 6.825 No effect 32 105 15 50.925 33.600 20.475 23 29.100 9.200 700 Total 97 64 9 200 Chi-Sq 13.938, DF 4, P.Value 0.007 (a) State the and alternative hypotheses for this test (b) Discuss whether the conditions for a chi-square inference procedure are met for tese data. (c) Given the results from the chi-square test, what should the advisory board conclude? (d) Based on your conclusion in part (c), which type of error (Type I or Type II) might the advisory board have made Describe this error in the context of the question. Q184 3:35 PM cise.ufl.edu I CLARO LTE Partial 001 101 1 100 +–Remainder The governing equation is as follows Dividend = Quotient . Divisor + Remainder. 3.3.2.1. Unsigned Division. The unsigned division algorithm that s similar to Booth’s algorithm is shown in Figure 3.19a, with an example shown in Figure 3.19b The ALU schematic diagram in given in Figure 3.19c. The analysis of the algorithm and circuit is very similar to the preceding discussion of Booth’s algorithm. M +-Divisor Q ? Dividend Count ? n, A +-0 START Shift left: A, Q AM Count ? Count-1 Quotient in Q Count 0 ENDRemainder in A A Q M 0011 0000 011 Initial values 0000 1110 Shift 1101 0000 1110 A-A+M 0001 1100 Shif 1110 0001 1100 A A+M 0011 1000 Shift A-A-M 2 (MMM) 1001 Q, 0001 0010 Shift 1110 0001 0010 A=A+M Divisor 4′ Add 2 Subtract 32-bit ALU 4′ write ( 4 write I Control 1 SLL Dividend Figure 3.19. Division of 32-bit Boolean number representations: (a) algorithm, (b) example using division of the unsigned integer 7 by the unsigned integer 3, and (c) 9 2
The Correct Answer and Explanation is :
(a) State the Null and Alternative Hypotheses for this Test
The null and alternative hypotheses for the chi-square test of association are:
- Null Hypothesis (H₀): There is no association between the number of hours worked per week and the perception of the effect of part-time work on academic achievement. In other words, the distribution of perceptions is the same across the different categories of hours worked (less than 1, 11 to 20, more than 20).
- Alternative Hypothesis (H₁): There is an association between the number of hours worked per week and the perception of the effect of part-time work on academic achievement. In other words, the distribution of perceptions differs across the different categories of hours worked.
(b) Discuss Whether the Conditions for a Chi-Square Inference Procedure Are Met
The conditions for performing a chi-square test are:
- Randomness: The data comes from a simple random sample of 200 students, as stated in the problem, which satisfies the randomness condition.
- Expected Frequency: The expected counts for each cell in the contingency table must be sufficiently large. Typically, each expected count should be at least 5. In the table, the expected counts are provided, and none are less than 5, so this condition is met.
- Independence: The observations must be independent of each other. Since the survey involves a simple random sample of students and there is no indication that the responses are related, this condition is also satisfied.
Since all conditions are met, the chi-square test is appropriate for this data.
(c) Given the Results from the Chi-Square Test, What Should the Advisory Board Conclude?
From the chi-square test output:
- Chi-Square Statistic (χ²) = 13.938
- Degrees of Freedom (df) = 4
- P-value = 0.007
The p-value of 0.007 is less than the commonly used significance level of 0.05. Therefore, we reject the null hypothesis. The advisory board should conclude that there is a significant association between the number of hours worked per week and the perception of the effect of part-time work on academic achievement.
(d) Based on Your Conclusion in Part (c), Which Type of Error Might the Advisory Board Have Made? Describe This Error in the Context of the Question.
Given that the null hypothesis was rejected, the advisory board might have made a Type I error if there was no actual association between the number of hours worked and academic achievement perceptions, but the test led to the conclusion that there was one. A Type I error occurs when a true null hypothesis is incorrectly rejected.
In the context of this problem, a Type I error would mean that the advisory board might believe that part-time work does affect academic achievement, when, in reality, there might be no significant relationship between them.