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Stat 151 Lab E02 Lab Assignment #4

a) The findings may be generalized to a population but only if we have a Simple Random Sample. Also, the test subjects must be a representative of the population that they were taken of to have the results generalized to the whole population. b) Since this is a random sample, only the age was a factor of the test subjects being chosen for this experiment. Because of this, we cannot use this study to assess the effect of age on posture control because of all the confounding variables that we do not know about. The test subjects could have been in accidents or had problems that left their posture “different” from the rest of the test subjects. c) This piece of information could have been useful in the study because the separation of time between the two trials could have altered the results. The test subjects, in the time given between the trials could have practiced or studied the task at hand and could have thought of ways to minimize the actual experiment time. Having this time to prepare for the experiment a second time because you have done it already could have given slightly off results. d) If there was a difference in swaying between the males and females, it would alter the mean to a smaller value or a larger value depending on the performance of the two sexes. However, since we do not know the results of the females, we cannot conclude what would happen exactly but rather that there would be a “skew” to the mean sway of the data for young adults. Gender is a confounding variable that we cannot account for that could alter our results. Question 2 a)

b) Forward and backward plane: Centers: The young center seems to be around 10 and the elderly center is around 25. Spreads: the spread for the young seems to be from 5 to 25 and the elderly spread is from around 18 to 30 but there is an outlier for the elderly at 50. Shapes: the sway for the elderly is much more than the sway of the young adults. young => large right skew Elderly => slight right skew (excluding outliers) Side to Side plane: Centers: The young center seems to be 15-ish and the elderly center seems to be 17-ish. Spreads : the spread for the young seems to stretch from 10 to about 23. And for the elderly, it seems to be from 12-ish to 24-ish but there are two outliers this time one at around 37-ish and 42-ish. Shapes : the sway of side by side is once again smaller in the young adults than in the elderly. Young => slight left skew Elderly => extreme right skew (excluding outliers). c)

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The plot does indicate a difference between the elderly and young in their ability to hold a stable upright position. According to the graph, the points for the young are more “concentrated” together than the elderly. The elderly points seem to be more spread and linear whereas the young points for side movements and fwd movements are closer together and not necessarily linear. The relationship between the two types of sways are in a fairly linear positions where the elderly have a more dependant relationship, as in, if they sway for fwd/back, chances are they will probably sway side/side. The young have either a more sway from side/side or a more sway of an fwd/back movement. Also, there are extreme cases for the elderly because there are outliers. Question 3 a) Summary statistics for Fwd/Back: Group by: AgeGroup Summary statistics for Side/Sid: Group by: AgeGroup b) Comparing the sway range between the two movements, we can conclude that for the fwd/back movements, the elderly and the young will sway more because the mean for their respective categories are larger. The extreme variance/ standard deviation values for the elderly tell us that there are outliers or values that should not be “there”. Once again, for the median values, the medians for the elderly and young in the fwd/back movement are larger than the side/side movement due to the fact that the two age groups sway more in the fwd/back position. AgeGroup Mean Variance Std. Dev. Median elderly 26.333334 95.5 9.77241 24 young 18.125 16.696428 4.0861263 17 AgeGroup Mean Variance Std. Dev. Median elderly 22.222221 105.44444 10.268615 18 young 15.125 15.267858 3.9074106 15

Question 4 a) Hypothesis test results: μ 1 : mean of Fwd/Back where AgeGroup = "elderly" μ 2 : mean of Fwd/Back where AgeGroup = "young" μ 1 - μ 2 : mean difference H 0 : μ 1 - μ 2 = 0 H A : μ 1 - μ 2 > 0 (without pooled variances) The hypotheses are: H 0 : μ 1 - μ 2 = 0 H A : μ 1 - μ 2 > 0 Test statistic: t-stat is 2.3034804 Distribution of test statistic: t-distribution with degrees of freedom 10.971389. P-value: 0.0209 Conclusion: p-value greater than or equal to 0.05 which means “there is not sufficient evidence to reject the null hypothesis.” P-value is less than 0.05 which means “we reject the null hypothesis.” With this data this means: p-value is greater than or equal to 0.05 means that we cannot say that mean Fwd/ Back sway for elderly is greater than mean fwd/ back sway for young. p-value less than 0.05 means we reject the null hypothesis, and conclude that mean fwd/ back sway for elderly is larger than mean fwd/ back sway for young. b) The assumptions for the test in part (a) are that the two samples are independently selected random samples and that the sample sizes are large so that the populations are approximately normal. The first of the assumptions are met in this case but for the sample size being approximately normal, that is not met because all the values and data are skewed. We only have 9 elderly and 8 young people so that is definitely not above 30 but since we have two trials for each person, which is closer to the normal but still not quite there. c) Hypothesis test results: μ 1 : mean of Side/Sid where AgeGroup = "elderly" μ 2 : mean of Side/Sid where AgeGroup = "young" μ 1 - μ 2 : mean difference H 0 : μ 1 - μ 2 = 0 H A : μ 1 - μ 2 > 0 (without pooled variances) Difference Sample Mean Std. Err. DF T-Stat P-value μ 1 - μ 2 8.208333 3.5634484 10.971389 2.3034804 0.0209

The hypotheses are: H 0 : μ 1 - μ 2 = 0 H A : μ 1 - μ 2 > 0 Test statistic: t-stat is 1.9227712 Distribution of the test statistic: t-distribution with degrees of freedom 10.500171 P-value: 0.041 Conclusion: P-value is less than 0.05 which means “we reject the null hypothesis.” Since the p-value is less than 0.05, we can reject the null hypothesis and conclude that the mean side/side sway for the elderly is larger than the mean side/side sway for the young. Question 5 a) Hypothesis test results: μ 1 - μ 2 : mean of the paired difference between Fwd/Back and Side/Sid H 0 : μ 1 - μ 2 = 0 H A : μ 1 - μ 2 > 0 The hypotheses are: H 0 : μ 1 - μ 2 = 0 H A : μ 1 - μ 2 > 0 Test statistic: t-stat is 2.2494452 Distribution of the test statistic: t-distribution with degrees of freedom 16 P-value: 0.0195 Conclusion: p-value is less than 0.05 which means “we reject the null hypothesis.” Since the p-value is less than 0.05, we can reject the null hypothesis and conclude that the mean of the paired difference between fwd/back is larger than side/side. b) 95% confidence interval results: μ 1 - μ 2 : mean of the paired difference between Fwd/Back and Side/Sid The interval is from (0.20663737, 6.9698334). Looking at the interval we see that the value of 0 is not in the interval therefore it is like the outcome of part (a) and saying that we reject the null hypothesis. It is not possible to get a value that is below the lower limit so that is why we can say it is like the hypothesis test results and rejecting the null hypothesis. Difference Sample Mean Std. Err. DF T-Stat P-value μ 1 - μ 2 7.0972223 3.6911423 10.500171 1.9227712 0.041 Difference Sample Diff. Std. Err. DF T-Stat P-value Fwd/Back - Side/Sid 3.5882354 1.5951647 16 2.2494452 0.0195 Difference Sample Diff. Std. Err. DF L. Lim U. Lim Fwd/Back - Side/Sid 3.5882354 1.5951647 16 0.20663737 6.9698334

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c) Hypothesis test results: μ 1 - μ 2 : mean of the paired difference between Fwd/Back and Side/Sid H 0 : μ 1 - μ 2 = 0 H A : μ 1 - μ 2 > 0 Group by: AgeGroup P-value for elderly: 0.0634 P-value for young: 0.1063 Conclusion: The p-values for the elderly and the young are 0.0634 and 0.1063 respectively, which are larger than 0.05. Because of this we cannot reject the null hypothesis and cannot conclude anything. Compared to part (a) and comparing the differences of the movements, we could reject the null hypothesis since it was smaller than 0.05 but here with the age group differences we cannot. AgeGroup Sample Diff. Std. Err. DF T-Stat P-value elderly 4.111111 2.4120326 8 1.7044177 0.0634 young 3 2.1876276 7 1.3713486 0.1063

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