Psychology

Section I
The data that will be used in this analysis is obtained from the 9th grade students performance and behavior data. The data consists of 216 observations from different variables regarding the performance and behavior of the students. The dependent variable that will be used for the analysis is the IQ scores of 9th grade students. The IQ score is an interval level of measurement. On the other hand, the English level of the students is the factor variableindependent variable that will be used for the analysis. English level is a nominal level of measurement.

Section II
One-way analysis of variance (ANOVA) will be used to determine if there is a difference between the average IQ scores of students among the English levels. In order to employ the analysis, there are certain assumptions needed to verify. The following are the assumptions of one-way ANOVA observations should be randomly sampled, the population variances should be homogeneous, the dependent variable (IQ scores) should be normally distributed and there should be no extreme outliers in the data (Howell, 2008).

First assumption to be tested is the random sampling. Random samples obtained in the observation depend on how the data are obtained from the participants of the 9th grade students performance and behavior data. As far as the survey data is concerned, researchers always try to obtain the data as random as possible. In case, the data is not randomly sampled, the analysis is still conducted for the sake of discussion.

Second assumption to be tested is the normality of the dependent variable. In order to test the normality of the dependent variable, the researcher will look at the summary statistics. In addition, histograms and Q-Q plots in order to evaluate the normality the data. Lastly, Shapiro-Wilks test will be used to determine whether the data came from a normal distribution.

Table 1
Descriptive Statistics
Descriptives9th Grade English LevelStatisticStd. ErrorIQ ScoreCollege PrepMean111.03451.98314Median112.0000Variance114.052Std. Deviation10.67953Skewness-.484.434Kurtosis-.333.845GeneralMean102.1623.96829Variance144.389Std.

Deviation12.01618Skewness-.199.195Kurtosis1.127.389RemedialMean95.62122.16124Variance154.141Std. Deviation12.41536Skewness.692.409Kurtosis.451.798 From the table, one can see that the mean IQ score for college preparatory, general and remedial English level are 111.0345, 102.1623 and 95.6212 respectively. The given values are the average IQ scores for each English level. The standard deviations for college preparatory, general and remedial English level are 10.67953, 12.01618 and 12.41536 respectively. The given values are the average spread of the IQ scores from the mean IQ score of each English level. The skewness for college preparatory IQ score is equal to -0.484. The value means that the IQ scores for college preparatory English level is slightly skewed to the left. The kurtosis for college preparatory IQ score is equal to -0.333. The value means that the IQ scores for college preparatory English level is platykurtic or slightly flat. On the other hand, the skewness for general English level IQ score is equal to -0.199. The value means that the IQ scores for general English level is slightly skewed to the left. The kurtosis for general English level IQ score is equal to 1.127. The value means that the IQ scores for general English level is leptokurtic or peaked. Lastly, the skewness for remedial IQ score is equal to 0.692. The value means that the IQ scores for remedial English level is slightly skewed to the right. The kurtosis for remedial IQ score is equal to 0.451. The value means that the IQ scores for remedial English level is leptokurtic or slightly peaked.

Figure 1. Histogram of College Preparatory English Level IQ Score.
From the figure, one can see that the data is skewed from the right. In addition, the data seems to be slightly flat because most of the data have almost the same frequency. Overall, the data does not seem to approximate normal distribution.

Figure 2. Histogram of General English Level IQ Score.
From the figure, one can see that the data is skewed to the left. In addition, the data is slightly peaked. Overall, the data does seem to approximate normal distribution.

Figure 3. Histogram of Remedial English Level IQ Score.
From the figure, one can see that the data is skewed to the right. In addition, the data is slightly peaked. Overall, the data does not seem to approximate normal distribution.

Figure 4. Normal Q-Q Plot of College Preparatory IQ Score
The figure shows the normal Q-Q plot of college preparatory IQ score. From the figure, one can see that most of the data points do not lie on the straight line. Only few of the data points are on the line or very near the line. There are also data points that are very far from the line which may be possible outliers. From the figure, one can conclude that there data does not approximate normal distribution.

Figure 5. Normal Q-Q Plot of General English Level IQ Score
The figure shows the normal Q-Q plot of general English Level IQ score. From the figure, one can see that most of the data points are on the straight line or are very near the straight line. Only few of the data points are not on the line and are very far from the line. There are also data points that are very far from the line which may be possible outliers. From the figure, one can conclude that the data does approximate normal distribution.

Figure 6. Normal Q-Q Plot of Remedial English Level IQ Score
The figure shows the normal Q-Q plot of remedial English Level IQ score. From the figure, one can see that most of the data points do not lie on the straight line. Only few of the data points are on the line or very near the line. There are also data points that are very far from the line which may be possible outliers. From the figure, one can conclude that the data does not approximate normal distribution.

In order to determine if the data comes from a normal distribution, Shapiro-Wilks test for normality is conducted. The null hypothesis to be tested is that the IQ scores for each 9th grade English level come from normal distribution. The test was conducted at 0.05 significance level. The decision is to reject the null hypothesis when the p-value of the test statistic is less than the significance level. Otherwise, fail to reject the null hypothesis. After conducting Shapiro-Wilks normality test, the following results were obtained.

Table 2
Test for Normality
Tests of Normality9th Grade English LevelKolmogorov-SmirnovaShapiro-WilkStatisticdfSig.StatisticdfSig.IQ Scoredimension1College Prep.08829.200.95629.257General.053154.200.987154.174Remedial.10733.200.96333.320a. Lilliefors Significance Correction. This is a lower bound of the true significance. From the table, one can see that the Shapiro-Wilks statistic for College preparatory, general and remedial level IQ score are equal to 0.956, 0.987 and 0.963 respectively. The values 0.257, 0.174 and 0.32 are the p-value of the said English level. Since the p-values of the 9th grade English level IQ scores are greater than 0.05, then the researcher failed to reject the null hypothesis. Thus, one can conclude that the IQ scores for each 9th grade English level come from a normal population. After evaluating the descriptive statistics, graphs and the test for normality, the researcher found out that the data does not approximate normal distribution.

In order to determine whether the data contains extreme scores, boxplots were created. The following is the boxplots for each 9th grade English level.

Figure 7. Boxplot of 9th Grade English Level IQ Score
The figure shows the boxplot for each of 9th grade English level. From the figure one can see that the boxplot for college preparatory does not have any extreme values. On the other hand, general and remedial English levels have extreme values.

In order to test for the homogeneity of the variances, Levenes test for Homogeneity of variance was conducted. The null hypothesis to be tested is that the variance of the IQ score is equal among the 9th grade English level. The test is conducted at 0.05 significance level. The decision is to reject the null hypothesis when the p-value of the test is less than the significance level. Otherwise, fail to reject the null hypothesis. After conducting Levenes test for homogeneity of variance, the following results were obtained.

Table 3
Levenes Test
Test of Homogeneity of VariancesIQ ScoreLevene Statisticdf1df2Sig..1402213.869 From the table, one can see that the Levene statistic is equal to 0.140 with a corresponding p-value of 0.869. Since the p-value of the test is greater than the significance level, then the researcher failed to reject the null hypothesis. Thus, the variance of the IQ score is equal among the 9th grade English level.

Most of the assumptions of One-way ANOVA were violated according to the results of the graphs and tests conducted. However, One-way analysis of variance will still be employed for the sake of discussion.

Section III
One-way analysis of variance is used to determine if there is a difference in the mean of three or more groups. The test uses the variance of the groups to compare each other. However, one-way ANOVA is only limited to determining if difference exist between groups. One-way ANOVA cannot detect where the difference lies on among the groups. In order to determine where the differences lie, post hoc analysis will also be conducted. The post hoc analysis to be used is the Least Significant Differences (LSD) test (Howell, 2008). In accordance with one-way ANOVA, the following hypothesis will be tested.

Null hypothesis There is no difference among the mean IQ scores of 9th grade English level.

Alternative There is at least one difference among the mean IQ scores of 9th grade English level.

The test will be conducted at 0.05 alpha level for both one-way ANOVA and the post hoc test.

Section IV
The decision is to reject the null hypothesis when the p-value of the F-statistic is less than the stated significance level of 0.05. Otherwise, the researcher will fail to reject the null hypothesis. After conducting one-way ANOVA, the following results were obtained.

Table 4
One-way ANOVA
ANOVAIQ ScoreSum of SquaresdfMean SquareFSig.Between Groups3686.73421843.36712.994.000Within Groups30217.422213141.866Total33904.156215
The table shows the result of one-way ANOVA test. From the table, one can see that the F-statistic is equal to 12.994 with a corresponding p-value of p0.001. Since the p-value of the F-statistic is less than the 0.05 significance level, the researcher rejects the null hypothesis. Thus, the researcher is 95 confident that there is at least one difference among the mean IQ score of 9th grade English level.

Figure 8. Means Plot of IQ score for each English Level

From the figure, one can see that the mean IQ scores for each grade level decreases from college preparatory up to the remedial English level. Thus, a difference really exists between the IQ scores of each English level.

In order to determine where the difference lies, least significant difference (LSD) test is conducted. The null hypothesis to be tested is that there is no difference between the mean IQ score for each pair of 9th grade English level. After conducting LSD test, the following results were obtained.

Table 5
Post Hoc Test
Multiple ComparisonsIQ Score
LSD(I) 9th Grade English Level(J) 9th Grade English LevelMean Difference (I-J)Std. ErrorSig.95 Confidence IntervalLower BoundUpper Bounddimension2College Prepdimension3General8.872152.41104.0004.119613.6247Remedial15.413273.03165.0009.437421.3891Generaldimension3College Prep-8.872152.41104.000-13.6247-4.1196Remedial6.541132.28477.0052.037511.0448Remedialdimension3College Prep-15.413273.03165.000-21.3891-9.4374General-6.541132.28477.005-11.0448-2.0375. The mean difference is significant at the 0.05 level. The table shows the post hoc test for the difference in the mean IQ score of 9th grade English level. From the table, one can see that the college preparatory English level differs in mean IQ score in both General and Remedial English level. The corresponding test values for the differences are 8.87215 and 15.41327 with a p-values less than 0.05. On the other hand, General also differs from Remedial English level. The corresponding test statistic for the difference is 6.54113 with a p-value of less than 0.05. Since the p-value of the test statistic is all less than 0.05, then the researcher reject the null hypothesis. Thus, there is a difference in mean IQ score for each pair of 9th grade English Level.