Slide 2 Notes
The research topic that will be addressed is of a hypothetical nature and therefore the data is also largely hypothetical. With that in mind, the comparison distribution is a distribution of means of the 44 pre-k classes in a particular school district. For the past year the school district has kept records of when children, age 60 months to 72 months, have learned their entire alphabet. The average of this distribution of means was 66.52 months and the standard deviation was 2.23. This age group was chosen to isolate cognitive development, dependent on age, from scholastic development, dependent on when the child started the grade. The point in keeping track of the age is to try and correlate teaching practices with particular programs of study. This year the school district decided to implement a statistical study to determine the effect of 30 minutes of extra-curricular (after school) reading for a pre-k class. The school district chose a class that had about average test scores, about average ethnicity, about average size class, and about average income-per-family. This particular class had 36 pre-k students between the ages of 60 months and 72 months. At the beginning of the year the parents of the students were asked if their children could stay after school each day to take part in a study about school curriculum. Every day after school the teacher would read out loud to the class from a book that emphasized learning letters. The class was asked to say the letter and point to the letter on every page. At the end of every month the class took a test to determine which students knew their complete alphabet. The test scores were recorded and tabulated at the end of the school year to determine if the extra reading was statistically significant.
Slide 3 Notes
The above scores are the means collected from the various pre-k classes in the districted for the last school year. As you can see the scores follow a normal distribution and is uni-modal. As far as kurtosis, the tails are neither heavy nor light, but largely consistent with a normal distribution. Before proposing a research hypothesis for the results of this study it would be prudent to look at how effective this study will have to be in order to have significant statistical power. The statistical power of a study is the, “…probability that the study will produce a statistically significant result if the research hypothesis is true” (Aron, Aron, & Coups, 2006, p. 197). In this case, a suitable power of .8 or 80% will suffice to demonstrate statistical power. In this study the power calculator on W. H. Freeman’s web site for biochemistry was used to calculate the score that would need to be reached by the study in order to satisfy a .08 statistical power (Power, n.d.). This score was 65.59 months. So for the study to reach a power of .8 the test group (n=36) must average at least 65.59 months old when they know the entire alphabet. This power calculated score would have an effect size of .42, indicating a small to medium effect size. The predicted effect size is reached by subtracting the mean of the sample group, as calculated at .8 statistical power, and dividing the answer by the standard deviation of the population: (66.52-65.59)/2.23.
Power will be .8 at 65.59 and effect size will be .42 at this mean…a small to medium effect size.
Slide Notes 4
Population 1 is represented as the one pre-k class that will receive 30 minutes of extracurricular reading every school day. Population 2 is the distribution of means of all of the pre-k classes in that particular district from the previous year. The null hypothesis is that there will be no significant difference found between population 1 and population 2 at the end of the year; that the mean of the sample will be equal to or greater than the mean score for all of the classes from the previous year. The alternative hypothesis, or research hypothesis, is that there will be a significant change in the number of months it takes for the children to learn the alphabet while being read to after school; that the mean of the sample will be less months than the mean score from the classes last year. This is a directionally low, one-tailed study, only concerned with decreased means. Therefore, any increase in months for the sample will result in statistical insignificance at the end of the study.
Slide 5 Notes
The comparison distribution is the distribution of means for all 44 pre-k classes from the previous year. Rule 1 of the distribution of means states that the distribution of means is equal to the population mean. Now, here we are talking about all 44 classes from last year. So, the mean of 66.52 stands as the mean for the comparison distribution because it actually does encompass the entire population, as long as we are only considering the implications for that one school district’s pre-k program. Rule 2a and 2b state that the variance for a distribution of means is equal to the variance of the population divided by the number of participants in each subsequent sample. Again, because the distribution that we already posses is both representative of a population and a distribution of means, the standard deviation stands at 2.23. Lastly, the third rule states that a distribution of means is normally shaped if there are more than 30 individuals in each sample and the distribution of the sample’s are normal. There are on average 36 students in each class with a very small variance, and each sample is normally distributed. Therefore, the shape of the distribution of means is approximately normal. In conclusion, the characteristics of the comparison distribution are a mean of 66.52, with a standard deviation of 2.23, and the distribution is normal in shape.
Slide 6 Notes
For the purposes of this study the test will be considered statistically significant at a p-value of .05. This means that the mean of the scores from the sample group will have to be in the 5% of the low tail in order for the test to be considered statistically significant. This percentage equates to a z-score of -1.64 or a raw score of 62.86 months. If the mean of the sample group, the group being read to after school, is less than 62.86 months, then we can reject the null hypothesis. If the mean of the sample is any more than 62.86, then we will have to reject the alternative hypothesis as statistically insignificant.
Slide 7 Notes
The sample distribution is positively skewed with a mode of 61, 62. The mean of the sample is 62.77 months and the standard deviation is 2.35, with a variance of 5.51. This is a z-score of -1.68 on the comparison distribution. This means that the results are statistically significant and we can reject the null hypothesis. The alternative hypothesis has been found to be statistically significant in this case.
Slide 8 Notes
It is not only important to know if a sample’s results are statistically significant to the distribution of means (p<.05), it is also important to know the effect size for that difference. You can equate the effect size by taking the mean of population 1 from the mean of population two and dividing it by the standard deviation of the sample. In this study, the number comes out to -1.59, which means that the mean of the sample is 1.59 standard deviations less than the mean of the comparison distribution. This is s very large effect size, with a statistical power over 1.00 or 100%. So, not only was the study found to be statistically significant, the effect size and power are such that the results of the study can be described as statistically effective and powerful.
Slide 9 Notes
Even though this study was hypothetical in nature, if such finding were to be demonstrated statistically evidence for after school reading programs would begin to accumulate. The study found that reading after school for 30 minutes each day with a sample class decreased the time that it took for the children to learn their entire alphabet by an average 3.74 months. The effect size between the comparison distribution and the sample distribution was -1.59, which is a very large effect. The study was statistically significant and the null hypothesis, that population 1 results would not vary significantly from population 2 data, was rejected in favor of the statistically significant alternative hypothesis.
Slide 10 Notes
Aron, A., Aron, E., & Coups, E. (2006). Statistics for psychology (4th ed.). Upper Saddle River, NJ: Pearson/Allyn Bacon.
Power. (n.d.). Retrieved March 11, 2009, from W.H. Freeman & Company Web site: http://bcs.whfreeman.com/ips4e/cat_010/applets/power_ips.html