Applications of Sampling Procedures and Alternatives - Assignment Example

Applications of Sampling Procedures and Alternatives Table of Contents Table of Contents 1 Descriptive Statistics 2 Hypothesis Test: Independent Samples t-Test 11 Effect Size (Cohen’s d) and Post-hoc Power Analysis of t-Test 12 Sample Size for Correlational Study 14 Hypothesis Test: Chi-square Test of Independence 16 Los Angeles School (Power and Sample Size Analysis Using G-Power and SPSS) This project will use G-Power in conjunction with SPSS to do some basic power analysis for different types of tests. The data set used for analysis is the Los Angeles Unified School District. Descriptive Statistics The sample consisted of 316 students. The boys and girls percentage were 48.7%, and 51.3% respectively, figure 1. Majority of the students were Hispanic (45.6%), figure 2. The ethnic percentage of other students was Native American (0.3%), Asian (7.9%), African-American (12.3%), white (31.0%), and Filipino (2.9%). The sample consisted of about equal students from each school Alfa and school Beta, figure 3. As can be seen in figure 4, the grade (mark) percentage received by the students in algebra was A (5.7%), B (18.0%), C (18.4%), D (22.5%), and fail (35.4%). As can be seen in figure 5, the grade percentage received by the students in spring math was A (17.1%), B (24.1%), C (30.7%), D (14.2%), and fail (13.9%). As can be seen in figure 6, the grade percentage received by the students in English fall of 1995 was A (14.6%), B (29.4%), C (28.2%), D (13.0%), and fail (14.9%). As can be seen in figure 7, the grade percentage received by the students in English fall of 1994 was A (11.7%), B (31.0%), C (32.0%), D (13.6%), and fail (11.7%). As can be seen in figure 8, the bilingual status of the students was about 37.0% native English speaker, 18.0% IFEP (Foreign language spoken in home but student tested English Proficient), 14.2% LEP (currently in bilingual program) and 30.7% RFEP (Formerly LEP but transitioned to English). As can be seen in figure 9, the proficiency in native language of the students was about 57.9% English (No other language—English-only spoken in home), 16.5% Proficient in Native Language), 16.5% FNC (Functional in Native Language), 6.3% LTD (Limited proficiency in Native Language), and 2.8% NON (Native Language spoken in home but student has no proficiency in Native Language). The average CTBS Math NCE score of the students was 48.75 (SD = 17.88). About half of the student received score less than 49. About 50% of the students received scores in-between 38 to 61. The minimum and maximum CTBS Math NCE score of the students was 1.01, and 98.99, respectively. The distribution of CTBS Math NCE score of the students was normal (Skewness = -0.05). It can be said with 95% confidence that the mean CTBS Math NCE score of the students was in-between 46.77 to 50.73. The average CTBS Language NCE score of the students was 50.06 (SD = 17.94). About half of the student received score less than 50. About 50% of the students received scores in-between 40 to 61. The minimum and maximum CTBS Language NCE score of the students was 1.01, and 98.99, respectively. The distribution of CTBS Language NCE score of the students was approximately normal (Skewness = -0.17). It can be said with 95% confidence that the mean CTBS Language NCE score of the students was in-between 48.08 to 52.05. The average CTBS Math percentile of the students was 48.42 (SD = 25.44). About half of the student received percentile less than 48. About 50% of the students received percentile in-between 28 to 70. The minimum and maximum CTBS Math percentile of the students was 1, and 99, respectively. The distribution of CTBS Math percentile of the students was normal (Skewness = -0.04). It can be said with 95% confidence that the mean CTBS Math percentile of the students was in-between 45.60 to 51.24. The average CTBS Language percentile of the students was 50.41 (SD = 25.32). About half of the student received percentile less than 50. About 50% of the students received percentile in-between 32 to 70. The minimum and maximum CTBS Language percentile of the students was 1, and 99, respectively. The distribution of CTBS Language percentile of the students was normal (Skewness = -0.05). It can be said with 95% confidence that the mean CTBS Language percentile of the students was in-between 47.61 to 53.22. The average number of days students present in school was 75 days (SD = 11.5). About half of the students were present in school more than 76 days. Most of the students were present in school on 86 days. About 50% of the students were present in school in-between 70 to 84 days. The distribution of number of days present in school of the students was left (negatively) skewed (Skewness = -1.75). The average number of days students absent from school was 6 days (SD = 7.5). About half of the students were absent from school less than 3 days. About 50% of the students were absent from school in-between 1 to 8 days. The distribution of number of days absent from school of the students was right (positively) skewed (Skewness = 2.26). Figure 1: Gender percentage of students Figure 2: Ethnicity percentage of students Figure 3: School percentage Figure 4: Students’ distribution of mark in algebra Figure 5: Students’ distribution of mark in spring math Figure 6: Students’ distribution of mark in English 95 Figure 7: Students’ distribution of mark in English 94 Figure 8: Bilingual status of students Figure 9: Students’ proficiency in native language Table 1 Descriptive Statistics CTBS Math NCE score CTBS Language NCE score CTBS Math percentile CTBS Language percentile number of days present number of days absent N Valid 316 316 316 316 316 316 Missing 0 0 0 0 0 0 Mean 48.75 50.06 48.42 50.41 74.66 5.81 Std. Error of Mean 1.01 1.01 1.43 1.42 0.65 0.42 95% Confidence Interval for Mean Lower 46.77 48.08 45.60 47.61 73.39 4.99 Upper 50.73 52.05 51.24 53.22 75.93 6.63 Median 48.94 50 48 50 76 3 Mode 56.99 71.83 63 85 86 0 Std. Deviation 17.88 17.94 25.44 25.32 11.47 7.45 Skewness -0.05 -0.17 -0.04 -0.05 -1.75 2.26 Range 97.99 97.99 98 98 73 45 Minimum 1.01 1.01 1 1 13 0 Maximum 98.99 98.99 99 99 86 45 Percentiles 25 37.73 40.15 28 32 70 1 75 61.04 61.04 70 70 84 8 Hypothesis Test: Independent Samples t-Test The research question examined was: Is there is a difference between boys and girls in their grades in the Spring Math Course? The hypotheses tested were: H0: µboys = µgirls H1: µboys ≠ µgirls The results of the test were statistically not significant, t(314) = 1.59, p = .113. Thus, the null hypothesis is retained; there is statistically no difference between boys and girls in their grades in the Spring Math Course. However, girl students (M = 2.27, SD = 1.27) reported greater mark in spring math as compared to boy students (M = 2.05, SD = 1.25). Table 2 Group Statistics gender N Mean Std. Deviation Std. Error Mean mark in spring math female 162 2.27 1.271 .100 male 154 2.05 1.254 .101 Table 3 Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper mark in spring math Equal variances assumed 1.150 .284 1.59 314 .113 .226 .142 -.053 .506 Equal variances not assumed 1.59 313.6 .112 .226 .142 -.053 .506 Effect Size (Cohen’s d) and Post-hoc Power Analysis of t-Test The pooled standard deviation of students’ grades in the Spring Math Course was about 1.263. The effect size (Cohen’s d) of students’ grades in the Spring Math Course was about 0.174. Figure 10: Post-hoc Power analysis for two-tailed t-test using G-Power The power for two-tailed test is 0.34. This means that if there were a real difference of this magnitude, there is only a 34% chance of finding it with a sample of this size. The value of t to reach to be significant for two-tailed test is 1.97 (critical value), however the test statistic t is 1.59. Figure 11: Post-hoc Power analysis for one-tailed t-test using G-Power The power for one-tailed test is 0.46. This means that if there were a real difference of this magnitude, there is only a 46% chance of finding it with a sample of this size. The value of t to reach to be significant for one-tailed test is 1.65 (critical value), however the test statistic t is 1.59. Sample Size for Correlational Study Suppose you were going to design a correlational study in which you couldn't assume that the effect size (r2) was more than .15, but you want to achieve 80% power with alpha = .05. Use G-Power to determine how large a total sample you would need to get such a level of power. Figure 12: Priori Sample size analysis for one-tailed correlation study using G-Power The sample size required for a one-tailed correlational study for the effect size (r2) no more than .15, achieving 80% power with alpha = .05 is 270. Figure 13: Priori Sample size analysis for two-tailed correlation study using G-Power The sample size required for a two-tailed correlational study for the effect size (r2) no more than .15, achieving 80% power with alpha = .05 is 343. Hypothesis Test: Chi-square Test of Independence The research question examined was: whether boys tend to be put into bilingual education more often than girls? The hypotheses tested were: H0: The gender and bilingual capability are independent. H1: The gender and bilingual capability are dependent The results of the test were statistically not significant, χ2(1, N = 316) = 1.37, p = .242. Thus, the null hypothesis is retained; the gender and bilingual capability are independent. In other words, there is no evidence to conclude that boys tend to be put into bilingual education more often than girls. Table 4 Gender * Bilingual Capability Crosstabulation Bilingual Capability Total no bilingual capability some bilingual capability gender female Count 65 97 162 Expected Count 60.0 102.0 162.0 male Count 52 102 154 Expected Count 57.0 97.0 154.0 Total Count 117 199 316 Expected Count 117.0 199.0 316.0 Table 5 Chi-Square Tests Value df Asymp. Sig. (2-sided) Exact Sig. (2-sided) Exact Sig. (1-sided) Pearson Chi-Square 1.368(b) 1 .242 Continuity Correction(a) 1.109 1 .292 Likelihood Ratio 1.371 1 .242 Fisher's Exact Test .247 .146 Linear-by-Linear Association 1.364 1 .243 N of Valid Cases 316 a Computed only for a 2x2 table b 0 cells (.0%) have expected count less than 5. The minimum expected count is 57.02. The effect size (w) of Chi-square test was about 0.174. Figure 14: Post-hoc Power analysis for Ch-square test using G-Power The post hoc power of Ch-square test to detect a true result is about 0.22. This means that if there were a real association between gender and bilingual capability, there is only a 22% chance of finding it with a sample of this size. The value of χ² to reach to be significant is 3.84 (critical value), however the test statistic χ² is 1.37. Figure 15: Priori Sample size analysis for Ch-square test using G-Power A sample of about 1757 would have been required to find significance in a relationship with the effect size of 0.07 detected in this sample achieving 80% power with alpha of .05. Read More