Statistics for the Behavioral and Social Sciences - Assignment Example

Essay We have totally 1000 participants grouped into two groups, with one group of 500 participants getting the shot and another group of 500 participants getting nasal spray. Out of the 500 people treated with the shot, 80 developed flu and 420 did not develop the flu. Again out of 500 participants treated with nasal spray, 120 developed the flu and 380 did not develop the flu. Clearly this is a test belongs to testing of two sample proportions for large samples. Null Hypothesis H0: The proportion of people developed flu with shot treatment is equal to the proportion of people developed flu with nasal spray treatment P1=P2. Alternative Hypothesis H1: The proportion of people developed flu with shot treatment is equal to the proportion of people developed flu with nasal spray treatment P1#P2. Level of Significance: 5% level or =0.05. Test Statistic or observed value: Zo= follows standard normal distribution with mean 0 and variance 1. The critical value of Z is Ze=1.96 at 5% level (two tailed test) and 2.58 at 1% level (two tailed). SE(p1–p2)= where p= and q=1–p In this problem n1=500, n2=500 p1=0.16, p2=0.24, p=(80+120)/1000=0.2, q=1–0.2=0.8 SE(p1–p2)=0.025298, therefore, Z0=–3.16 |Zo|=3.16 with probability 0.00078 approx.0.0008. Conclusion: Since the observed value of Z ie.Zo (3.16) ≥Ze (1.96) with probability 0.0008, we reject the null hypothesis H0 and accept alternative hypothesis H1 and conclude that there is a highly significant difference between the proportion of people with flu treated by shot and the proportion of people with flu treated by nasal spray. Justification: The sample size is 500 from each sample is well sufficient for the study, as the normal test insists that the sample size be minimum 30. Thus, there is no controversy in the sample size. As far as the test procedure is concerned there are no limitations in the study except sampling fluctuations. The same test can be done using chi square test for association of attributes. For this we categorize the people affected with flu and non flu as one attribute and the treatments with shot and nasal spray as another attribute. Null Hypothesis H0: There is no association between the type of treatment and causation of flu. Alternative Hypothesis H1: There is a association between the type of treatment and causation of flu. Level of significance: 5% level or =0.05. Test statistic: 02= follows chi square distribution with 1 df Contingency table Affected with flu Row Total Yes No Type of treatment Shot a=80 b=420 r1=500 Nasal spray c=120 d=380 r2=500 Column Total c1=200 c2=800 N=1000 With the use of the above contingency table, after computation of 02, using the formula , the chi square observed value (test statistic) is found to be 10.00. The expected value of chi square is 3.84 with 1 d.f at 5% level of significance. Conclusion: Since the observed value of chi square is far above than the expected value of chi square for 1 d.f. at 5% level, we reject the null hypothesis H0 and accept the alternative hypothesis H1 and conclude that there is an association between the type of treatment and causation of flu. Essay 2 The chosen variables are IQ (Intelligent Quotient, scaling variable) and GPA, Grade Point Average, scaling variable). The statistical tests that could be used are the test for significance of correlation using Student’s t-test. The test is given below: Null Hypothesis H0: The observed correlation is not significant. Alternative Hypothesis H1: The observed correlation is significant. Level of Significance: 5% level or =0.05. Test Statistic: to= which follows t distribution with n–2 d.f. the correlation coefficient is r=0.75 and the sample size is 18. The observed t value is to=6. The expected value is te=2.12 for 16 d.f. at 5% level. Since the observed value (to) is greater than the expected value (te) with probability of significance 0.0000185, we reject the null hypothesis Ho and accept the alternative hypothesis H1 and conclude that the given correlation coefficient is highly significant. The correlation is strong, since it is above +0.5 despite the small sample size of 18, and the correlation is also positive. There is a direct relationship between Intelligent Quotient (IQ) and Grade Point Average (GPA), and so this correlation implies to a considerable extent that individuals with high Intelligence Quotients (IQ) have high Grade Point Averages (GPA). In addition, this correlation also provides evidence that high IQ causes GPA to go higher. Correlation is just a measure of inter relationship between two variables whereas causation is a functional relationship between two variables (one influencing the other). Sample Size is a key factor that affects the size of this correlation. Importantly, yes, correlation a good test for predicting GPA. Essay 3 The above analysis was run with MS Excel. The following results were obtained: Descriptive Statistics of Reaction time Mean 6.36 Standard Error 0.758995 Median 6.05 Mode 9.5 Standard Deviation 3.39433 Sample Variance 11.52147 Kurtosis 0.578947 Skewness 0.775233 Range 13 Minimum 2.2 Maximum 15.2 Sum 127.2 Count 20 Confidence level (95%) 1.588595 The above table shows that the mean reaction time is 6.36 with Standard Deviation 3.39 and the standard error being 0.76. The 95% confidence interval is 6.36+1.59 ie.(4.77,7.95). Answer: The data is arranged in ascending order and divided into two groups each containing 10 values. I sample (low reaction) 2.2 2.5 2.7 2.9 3.1 3.5 4.1 4.3 4.7 4.8 II sample (high reaction) 7.3 7.6 8.1 8.2 8.5 9.2 9.3 9.5 9.5 15.2 The descriptive statistics for each group is given below: Descriptive Statistics: Lower reaction time Mean 3.480 95% Confidence Interval for Mean Lower Bound 2.807 Upper Bound 4.153 Standard Error 0.2977 Median 3.300 Variance .886 Std. Deviation .9414 Minimum 2.2 Maximum 4.8 Range 2.6 Interquartile Range 1.8 Skewness .190 Kurtosis -1.571 Interpretation: From the above table, it is observed that mean of the lower reaction group is 3.48 with standard deviation 0.9414 and standard error 0.2977. The 95% confidence interval is (2.807,4.153). The outliers are 2.2, 2.5, 2.7, 4.3, 4.7, 4.8. This shows that the outliers are above 50% which shows the data has high variability and if the sample size is increased, this may lead to the variability less. Descriptive Statistics: Higher reaction time Mean 9.240 95% Confidence Interval for Mean Lower Bound 7.640 Upper Bound 10.840 Standard Error 0.7075 Median 8.850 Variance 5.005 Std. Deviation 2.2372 Minimum 7.3 Maximum 15.2 Range 7.9 Interquartile Range 1.5 Skewness 2.444 Kurtosis 6.869 Interpretation: From the above table, it is that mean of the higher reaction group is 9.24 with standard deviation 2.2372 and standard error 0.7075. The 95% confidence interval is (7.64,10.84). The outliers are 7.3, 7.6 and 15.2. This shows that the outliers are 30% which shows the data has less variability and if the value 15.2 is excluded; all the data values will be within the 95% confidence interval. However if sample size is increased, this may lead to the variability less. From the above figure it is clear that there is one outlier which is the value 15.2. Descriptive Statistics: Lower reaction time Mean 3.480 95% Confidence Interval for Mean Lower Bound 3.051 Upper Bound 3.909 Standard Error 3.300 Median .840 Variance .9163 Std. Deviation 2.2 Minimum 4.8 Maximum 2.6 Range 1.6 Interquartile Range .174 Skewness -1.497 Kurtosis 3.300 Interpretation: From the above table, it is observed that mean of the lower reaction group is 3.48 with standard deviation 0.9163 (slightly less than the previous one) and standard error 0.2049 (31% drastical reduction). The 95% confidence interval has shrinked closer because the standard error has reduced drastically and the Confidence interval is (3.051, 3.909). The outliers are 2.2, 2.5, 2.7, 2.9, 4.1, 4.3, 4.7, 4.8 all two times. This shows that the outliers are above 75% which shows the data has high variability and even if the sample size is increased, this caused the outliers further more. Descriptive Statistics: Higher reaction time Mean 9.240 95% Confidence Interval for Mean Lower Bound 8.221 Upper Bound 10.259 Standard Error .4869 Median 8.850 Variance 4.741 Std. Deviation 2.1775 Minimum 7.3 Maximum 15.2 Range 7.9 Interquartile Range 1.4 Skewness 2.232 Kurtosis 4.728 Interpretation: From the above table, it is that mean of the higher reaction group is 9.24 with standard deviation 2.1775 and standard error 0.4869. The 95% confidence interval is (8.221,10.259). The outliers are 7.3, 7.6, 8.1, 8.2 and 15.2 two times. This shows that the outliers are 50% which shows the data has more variability than the data was expressed with only one time entries and still if 15.2 is excluded, all the data values may come within the 95% confidence interval. The effect is there is no change in the mean but there is a change in Standard Deviation, Standard error and confidence interval if the values are repeated twice. However, if the sample size is increased with no repetition but random values chosen, this may lead to the variability less. Read More