**Transcription of Educational Website Content:**---**Do students perform worse when they take an exam alone than when they take an exam in a classroom setting?** Eight students were given two tests of equal difficulty. They took one test in a solitary room and they took the other in a room filled with other students. The results are shown below.**Exam Scores**| | Alone | Classroom ||--------|-------|-----------|| Student 1 | 73 | 78 || Student 2 | 73 | 76 || Student 3 | 87 | 93 || Student 4 | 72 | 72 || Student 5 | 74 | 81 || Student 6 | 76 | 84 || Student 7 | 85 | 83 || Student 8 | 80 | 85 |Assume a Normal distribution. What can be concluded at the α = 0.10 level of significance?For this study, we should use [Select an answer]a. The null and alternative hypotheses would be:- \(H_0:\) [Select an answer] [Select an answer] [Select an answer] [please enter a decimal]- \(H_1:\) [Select an answer] [Select an answer] [Select an answer] [please enter a decimal]b. The test statistic \(t =\) [ ? ] = ______ (please show your answer to 3 decimal places)c. The p-value = ________ (Please show your answer to 4 decimal places)d. The p-value is [ ? ] [ < or > ] αe. Based on this, we should [Select an answer] the null hypothesis.f. Thus, the final conclusion is that …- ( ) The results are statistically significant at α = 0.10, so there is sufficient evidence to conclude that the eight students scored lower on average taking the exam alone compared to the classroom setting.- ( ) The results are statistically significant at α = 0.10, so there is sufficient evidence to conclude that the population mean test score taking the exam alone is less than the population mean test score taking the exam in a classroom setting.- ( ) The results are statistically insignificant at α = 0.10, so there is insufficient evidence to conclude that the population mean test score taking

Paired t-test is used to test the mean significant difference between two population means (dependent groups). The test statistic is given as follows t=∑dn∑d2-∑d2n-1 where d is the difference of paired value n is the sample size For this study, we should use Paired t-test (dependent t-test). Choose the paired t-test study (dependent t test) if you two measurements on the same subject (i.e., person, thing etc.). Thus, in this problem exam was conducted with same eight students but at different places. So, paired t-test is suitable method. a. Making of Hypotheses: Null hypothesis H0: μ1 - μ2 = 0 i.e., The population mean test score of taking the exam alone is equal to the population mean test score taking the exam in a classroom setting. Alternative hypothesis H1 : μ1 - μ2 < 0 i.e., The population mean test score of taking the exam alone is less than the population mean test score taking the exam in a classroom setting. Hence, the null and alternative would be: H0: μ1 - μ2 = 0 H1 : μ1 - μ2 < 0 b. Finding of test statistic: Attached table is as follows below Alone (x) Classroom (y) d = x-y d2 = (x-y)2 73 78 -5 25 73 76 -3 9 87 93 -6 36 72 72 0 0 74 81 -7 49 76 84 -8 64 85 83 2 4 80 85 -5 25 ∑d = -32 ∑d2 = 212 From the above table, we have ∑d = -32, ∑d2 = 212 and n =8 Then, the test statistic is t=-328212--3228-1 =-321696-10247 =-326727 =-3296 =-329.798t=-3.266 Hence, the test statistic is -3.266 c. Calculation of p-value: If test statistic is -3.266 with (n-1) = 7 df at 0.10 level of significance then p-value is 0.0069 [refer from one tailed t-test table values] Hence, the p-value is 0.0069 d. The p-value is less than (<) α i.e., p-value (0.0069) < α = 0.10 Decision rule for p-value: Reject null hypothesis if p-value is less than (<) 0.10. Otherwise, fail to reject the null hypothesis. Since, p-value (0.0069) < 0.10. So, we have sufficient evidence to reject null hypothesis at 0.10 level of significance. e. Based on this, we should reject the null hypothesis. f. Thus, the final conclusion is that, the results are statistically significant at α = 0.10, so there is sufficient evidence to conclude that the population mean test score taking the exam alone is less than the population mean test score taking the exam in a classroom setting. Hence, option. b is correct.