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Assessment 2: Hypothesis Testing, Correlation, Chi-Square, Dependent and Independent Samples T-Test. Section A: Short Answer Questions Question 1 In the chi-square test, when the expected value is less than 5, it implies that the assumption of the test is violated. The chi-square assumes that the expected value for the frequencies is more than 5. An expected value of 5 compromised the validity and reliability of the results. When the expected value is less than 5, the following should be done; 1. A different statistical test should be used since the assumption of the chi-square test had been violated. The test will take into account the type of the variable and the objective of the test. 2. The sample size should be increased to obtain a higher expected value. This will be collecting more data for the study. 3. The adjusted cells with a low expected value should be combined to create larger groups. The categories will be collapsed to increase the expected count of cells. 4. Corrections such as Yates' continuity correction for 2*2 tables should increase the accuracy when using a small expected value. Question 2 In this case, the most appropriate inferential statistical technique is correlation analysis. It examines the relationship between the Beck's Depression Inventory (BDI-II) scores and Drug Abuse Screening Test (DAST-10) scores among the inmates. The Beck's Depression Inventory (BDI-II) scores examine the level of depression among inmates, while the Drug Abuse Screening Test (DAST-10) scores measure drug use. The two variables are continuous with an interval level of measurement. Hence, the Pearson correlation coefficient will be used to assess the strength and direction of the relationship. Question 3 The power of a test is the probability of rejecting the null hypothesis when it is false. A test with a higher power is regarded as very sensitive and better at detecting the true effects. The power of the test has an indirect relationship with Type II error. Type II error (β) arises when we fail to reject the null hypothesis when it is false. A higher power of test decreased the probability of making Type II errors. The power of the test is denoted by (1- β) while Type II error (β). Lowering the power of the test will increase the probability of committing Type II errors. Increasing the power of the test will decrease the probability of committing Type II errors. The following actions will be implemented to have adequate power in a study. 1. Increasing the sample size. An increase in the sample size will increase the test's power by decreasing the probability of committing type II errors.

2. Increasing the effect size. An increase in the effect size will increase the test's power and reduce the likelihood of making Type II errors. Question 4 There is a positive relationship between statistical significance, sample size and the effect size. It implies that a larger sample size will increase a test's statistical significance and power, making the test more likely to detect the true effect. Hence, a larger effect size increases the statistical significance, showing a strong relationship linking the variables. Also, increasing the sample size makes it easier to detect smaller effect sizes. This suggests that a larger sample size allows for greater precision in estimating the true effect size. Question 5 There is a difference between the chi-square test and the parametric tests. The two differ based on the hypotheses, data, and assumptions. Hypotheses Chi-square tests are used to test hypotheses about the distribution of categorical data. On the other hand, parametric tests are used to test hypotheses about continuous data's means, differences or relationships. Data The chi-square tests categorical variables that are either ordinal or nominal. Besides, the parametric tests assess continuous variables measured on an interval or ratio scale. Assumptions For the chi-square tests, data is not normally distributed. It violates the assumption of normality. Besides, parametric tests assume normality of data and homogeneity of variance. Hence, it meets the assumption of normality and equality of variance. Section B: Calculations Question 1 a. State the null and alternate hypotheses The following null and alternative hypotheses will be tested. Null hypothesis (H0): There is no relationship between employment status and anxiety level. Alternative hypothesis (Ha): There is a relationship between employment status and anxiety level.

b. State the design requirements and assumptions needed to run this test Design Requirements and Assumptions i) The variables should be categorical. ii) The expected frequency for each cell should be at least 5. iii) The sample size should be large. iv) Data is drawn from a random sample. c. State the decision rule The decision rule for the chi-square test is the critical value. The critical value is compared with the test statistic; if the critical value is less than the test statistic, we reject the null hypothesis. We fail to reject the null hypothesis if the critical value is greater than the test statistic. d. Calculate the value of the test statistic and make a decision The test statistic is computed using the following formula. Employment Status Anxiety Level Low Median High Total Unemployed 13 (32.92) 40 (50.46) 67 (33.85) 120 Employed 68 (49.38)

90 (75.69) 22 (50.77) 180 Retired 26 (24.69) 34 (37.85) 30 (25.38) 90 Total 107 164 110 390 (13-32.92)2/32.92 + (68-49.38)2/49.38 + (26-24.69)2/24.69 + (40-50.46)2/50.46+ (90- 75.69)2/75.69 +(34-37.85)2/37.85 + (67-33.85)2/33.85 + (22-50.77)2/50.77 + (30-25.38)2/25.38 = 12.05 + 7.02 +0.07+2.17+2.71+0.39+32.46+16.30+0.84 = 74.02 Degrees of freedom, df = (r-1)(c-1) = (3-1)(3-1) = 4 Using the chi-square table, we find the critical value, alpha =0.01/2 and df = 4. Critical value = 14.8603 Since the critical value is less than the test statistic, , we reject the null hypothesis. e. State the conclusion Since the critical value is less than the test statistic, there is sufficient evidence to support the claim. Therefore, there is a significant relationship between employment status and anxiety level. f. Calculate and interpret the effect size, if appropriate

We calculate Cramer's V, which is the effect size of the chi-square test. V = = = = 0.31 This is a moderate effect size. It implies that there is a moderate relationship between employment status and anxiety level. Question 2 a. State the null and alternate hypotheses Null hypothesis (H0): There is no significant difference in daily water usage before and after the campaign. Alternative hypothesis (Ha): There is a significant difference in daily water usage before and after the campaign. b. State the design requirements and assumptions needed to run this test Design Requirements and Assumptions i) The data should be paired or matched ii) The data should be normally distributed. iii) The data should be randomly sampled from a population. c. State the decision rule The critical t-value will be compared with the test statistic (t-value). If the critical value is less than the test statistic, we reject the null hypotheses and decide there is a significant difference. We fail to reject the null hypothesis if the critical value is greater than the test statistic. Hence, there is no significant difference. The critical value will be tabulated at alpha =0.05/2 and df=n- 1= 7. d. Calculate the value of the test statistic and make a decision

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