Psychology

Section I
The data is obtained from 9th grade students  performance and behavior data. The data consists of 216 observations that will be used for the analysis. The variable of interest for the analysis is the IQ scores of the 9th grade students.

Section II
One sample t-test will be used to determine whether the average IQ score of the sample differs from the population average IQ score of 100. In order to conduct one sample t-test, one must first summarize and check certain assumptions regarding the data. The assumptions that are needed to be checked are normality of the data, random samples, sample size should be less than 30 and the population mean is known. In order to summarize the data, descriptive statistics will be used. On the other hand, normality of the data will be determined by using descriptive statistics through some graphs and the Shapiro-Wilks W test.

Table 1
Descriptive Statistics
DescriptivesStatisticStd. ErrorIQ ScoreMean102.3542.8544495 Confidence Interval for MeanLower Bound100.6700Upper Bound104.03835 Trimmed Mean102.3549Median103.0000Variance157.694Std. Deviation12.55762Minimum55.00Maximum137.00Range82.00Interquartile Range16.88Skewness-.114.166Kurtosis.333.330
The table shows the summary statistics for the IQ scores. From the table, one can see that the mean IQ score is equal to 102.3542. The value means that the average IQ score of 9th grade students is equal to 102.3542. The standard deviation has a value of 12.55762 from the table, which suggests that there is an average of 12.55762 spread of IQ score from the mean. The skewness of the IQ score is equal to -0.114, which means that the data is skewed slightly to the left. On the other hand, kurtosis is equal to 0.333, suggesting that the distribution of the data is slightly peaked.

The following figure shows the graph that will be used to determine if the data follows a normal distribution.

Figure 1. Histogram of IQ scores
From the figure, one can see that the data has some observations which are extremely small and extremely large as seen from the tails of the histogram. In terms of skewness, the said observations make the data slightly skewed to the left. The histogram is also slightly peaked due to the existence of very high frequency at the middle of the histogram. The data seems to approximate normality based on the histogram.

Figure 2. Normal Q-Q Plot of IQ Score.
From the figure, one can see that most of the data fall on the line. While some of the data are exactly on the line, they are nevertheless near the line. However, there are some data points which are very far away from the pile of data. The data points that are away from the data can be possible outliers. Overall, the data seems to approximately follow normal distribution.

In order to verify the normality assumption given in the figures, Shapiro-Wilks W test for normality will be conducted. The null hypothesis to be tested is that the IQ scores come from a 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 is less than 0.05. Otherwise, the researcher will fail to reject the null hypothesis. After conducting Shapiro-Wilks W test for normality, the following results were obtained.

Table 2
Shapiro-Wilks Test for Normality
Tests of NormalityKolmogorov-SmirnovaShapiro-WilkStatisticdfSig.StatisticdfSig.IQ Score.040216.200.994216.569a. Lilliefors Significance Correction. This is a lower bound of the true significance. From the figure, one can see that the Shapiro-Wilk W statistic is equal to 0.994. The statistic has a corresponding p-value of 0.569. Since the p-value of the test is greater than the significance level, the researcher failed to reject the null hypothesis. Thus, the IQ scores come from a normal distribution.

Some of the assumptions of one-sample t-test were violated. One of the assumptions that were violated is the assumption of sample size. The sample size is equal to 216 observations. The needed sample size to conduct the test should be less than 30.

Section III
In order to use one-sample T-test, one must establish the hypothesis on which the test will be conducted. One-sample T-test is used to test whether the mean of a sample is different from the mean of a population. One-sample t-test is usually conducted when the population mean is known, but the scores are not converted to standard scores (Howell, 2008). The following are the hypotheses that will be used for the test.

Null hypothesis The average IQ scores of 9th grade students is equal to the population mean IQ score of 100.
Alternative The average IQ scores of 9th grade students is not equal to the population mean IQ score of 100.
The hypothesis will be tested with the use of one-sample T-test. The test will be conducted at two-tailed test with an alpha level of 0.05.

Section IV
The decision is to reject the null hypothesis when the p-value of the t-statistic is less than the 0.05 significance level. Otherwise, the researcher will fail to reject the null hypothesis that the average IQ score 9th grade students is equal to the population mean IQ score of 100. After conducting one-sample T-test, the following results were obtained.
Table 3
One-sample T-test
One-Sample TestTest Value  100                                     tdfSig. (2-tailed)Mean Difference95 Confidence Interval of the DifferenceLowerUpperIQ Score2.755215.0062.35417.67004.0383

The table shows the result of the one-sample T-test. From the table, one can see that the t-statistic is equal to 2.755 with 215 degrees of freedom. The t-statistic has a corresponding p-value of 0.006. Since the p-value of the t-statistic is less than the 0.05 significance level, then the null hypothesis is rejected.

Thus, the results showed that there is a significant difference between the average IQ score of 9th grade students and the population IQ score of 100 (t  2.755, p  0.006). There are certain limitations regarding the conclusions obtained. Since the test was a two-tailed test, one cannot conclude that the IQ score of the sample is greater than or less than the population mean IQ score of 100. Another limitation of the conclusion is that there is a greater chance to reject the null hypothesis since the sample size is way too large for a one-sample T-test to be conducted. In one-sample T-test, the sample size of concern should be less than 30. In addition, the existence of possible outliers (extreme values) may have affected the one-sample test which in turn may have affected the conclusion obtained from the test.