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In an investigation of the visual scanning behavior of deaf children, measurements of eye movement were taken on nine deaf and nine hearing children. The table gives the eye movement rates and their ranks (in parentheses). Does it appear that the distributions of eye-movement rates for deaf children and hearing children differ? $$ \begin{array}{llc} \hline & \text { Deaf Children } & \text { Hearing Children } \\ \hline & 2.75(15) & .89(1) \\ & 2.14(11) & 1.43(7) \\ & 3.23(18) & 1.06(4) \\ & 2.07(10) & 1.01(3) \\ & 2.49(14) & .94(2) \\ & 2.18(12) & 1.79(8) \\ & 3.16(17) & 1.12(5.5) \\ & 2.93(16) & 2.01(9) \\ & 2.20(13) & 1.12(5.5) \\ \hline \text { Rank Sum } & 126 & 45 & \\ \hline \end{array} $$

Short Answer

Expert verified
Answer: Yes, the distributions of eye-movement rates for deaf children and hearing children differ significantly.

Step by step solution

01

Identify the sample sizes and rank sums

In this case, we have two groups: deaf children and hearing children, with nine observations in each group. The table gives the rank sum (\(R\)) for each group, which are \(R_1 = 126\) for deaf children and \(R_2 = 45\) for hearing children.
02

Calculate the Mann-Whitney U statistic

We can calculate the U statistic using the following equation: $$ U = n_1n_2 + \frac{n_1(n_1+1)}{2} - R_1 $$ Where \(n_1\) and \(n_2\) are the number of observations in group 1 (deaf children) and group 2 (hearing children) respectively, and \(R_1\) is the rank sum for group 1. In our case, \(n_1 = n_2 = 9\) and \(R_1 = 126\). Plugging these values into the equation, we get: $$ U = 9 \cdot 9 + \frac{9 \cdot (9+1)}{2} - 126 = 81 + \frac{9 \cdot 10}{2} - 126 = 81 + 45 - 126 = 0 $$ The U statistic is equal to 0.
03

Consult the Mann-Whitney U Test table

A Mann-Whitney U test table can be found in most statistics textbooks or online. Given our sample sizes of \(n_1 = n_2 = 9\) and our U statistic of 0, we can determine the critical value for U at a desired significance level. With a significance level of 0.05 (5%), the critical value for U is 17.
04

Compare the U statistic to the critical value

We compare our calculated U statistic (0) to the critical value (17) at the 0.05 significance level: $$ U = 0 < 17 = U_{critical} $$ Since our U statistic is smaller than the critical value, we reject the null hypothesis (which states that there is no difference between the two groups). This means that the distributions of eye-movement rates for deaf children and hearing children do appear to differ significantly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eye Movement Rates in Children
Eye movement rates in children play a crucial role in understanding their visual scanning abilities. These rates are indicators of how quickly and effectively children can shift their focus from one visual target to another. It's essential in contexts like reading, object tracking, and information processing.

Research comparing eye movement rates between different groups of children, such as those who are deaf and those with typical hearing, can reveal insights into the developmental impacts of auditory deprivation on visual processing. In the exercise, by comparing the eye movement rates of deaf and hearing children, we're looking to understand if there is a significant difference in visual scanning behavior driven by the auditory experiences, or lack thereof.

In educational settings, understanding these differences can help tailor teaching methods and tools to support visual learning processes more effectively, ensuring that all children, regardless of hearing ability, have the optimal opportunity to learn and interact with their environment.
Non-parametric Statistical Tests
Non-parametric statistical tests are useful when the data does not necessarily fit the assumptions required for parametric tests. One common assumption for many parametric tests is that the data follows a normal distribution. When this condition is not met, or when dealing with ordinal data or ranks rather than measurement data, non-parametric tests like the Mann-Whitney U Test become invaluable.

The Mann-Whitney U Test, specifically, determines whether there are significant differences between two independent groups. It does this by comparing the ranks of the data rather than the data itself, which makes it less sensitive to outliers and skewed distributions.

This approach is particularly beneficial in educational research, where diverse populations and unequal variances could potentially invalidate the assumptions of parametric tests. By employing a non-parametric test like the Mann-Whitney U, educators and researchers can make meaningful comparisons between groups with fewer concerns about the underlying data distribution.
Visual Scanning Behavior
Visual scanning behavior is the pattern and ability with which an individual moves their eyes to take in visual information. This encompasses how eyes are directed from one point to another and can indicate cognitive processes such as attention and memory recall.

In the context of the exercise, we are looking at visual scanning behavior as it pertains to rates of eye movement, specifically how deaf children's scanning behavior might differ from that of hearing children. Variations in visual scanning behavior can suggest differences in how individuals process visual stimuli and how they might compensate for sensory deficits.

Studies on visual scanning behavior can have practical applications in creating educational tools and environments. For children with hearing impairments, for example, enhanced visual cues could be integrated into classroom designs to support their learning experiences, promoting better educational outcomes through a better understanding of their visual scanning behaviors.

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Most popular questions from this chapter

To compare the effects of three toxic chemicals, \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C},\) on the skin of rats, 2 -centimeter-side squares of skin were treated with the chemicals and then scored from 0 to 10 depending on the degree of irritation. Three adjacent 2-centimeter-side squares were marked on the backs of eight rats, and each of the three chemicals was applied to each rat. Thus, the experiment was blocked on rats to eliminate the variation in skin sensitivity from rat to rat. a. Do the data provide sufficient evidence to indicate a difference in the toxic effects of the three chemicals? Test using the Friedman \(F_{r}\) -test with \(\alpha=.05 .\) b. Find the approximate \(p\) -value for the test and interpret it.

Give the null and alternative hypotheses, determine the degrees of freedom, find the appropriate rejection region with \(\alpha=.05\) and draw the appropriate conclusions. $$ T_{1}=35, T_{2}=63, T_{3}=22, n_{l}=n_{2}=n_{3}=5 $$.

Use the information given in Exercises \(4-7\) to calculate Spearman's rank correlation coefficient, where \(x_{i}\) and \(y_{i}\) are the ranks of the ith pair of observations and \(d_{i}=x_{i}-y_{i} .\) Assume that there are no ties in the ranks. \(d_{i}=\\{-6,-3,-3,-4,2,5,5,4\\}\)

A high school principal formed a review board consisting of five teachers who were asked to interview 12 applicants for a vacant teaching position and rank them in order of merit. Seven of the applicants held college degrees but had limited teaching experience. Of the remaining five applicants, all had college degrees and substantial experience. The review board's rankings are given in the table. $$ \begin{array}{cc} \hline \text { Limited Experience } & \text { Substantial Experience } \\ \hline 4 & 1 \\ 6 & 2 \\ 7 & 3 \\ 9 & 5 \\ 10 & 8 \\ 11 & \\ 12 & \\ \hline \end{array} $$ Do these rankings indicate that the review board considers experience a prime factor in the selection of the best candidates? Test using \(\alpha=.05 .\)

Competitive Running Is the number of years of competitive running experience related to a runner's distance running performance? The data on nine runners, obtained from a study by Scott Powers and colleagues, are shown in the table: $$\begin{array}{crc}\hline & \text { Years of Competitive } & \text { 10-Kilometer Finish } \\\\\text { Runner } & \text { Running } & \text { Time (minutes) } \\\\\hline 1 & 9 & 33.15 \\\2 & 13 & 33.33 \\\3 & 5 & 33.50 \\\4 & 7 & 33.55 \\\5 & 12 & 33.73 \\\6 & 6 & 33.86 \\\7 & 4 &33.90 \\\8 & 5 & 34.15 \\\9 & 3 & 34.90 \\\\\hline\end{array}$$ a. Calculate the rank correlation coefficient between years of competitive running and a runner's finish time in the 10 -kilometer race. b. Do the data provide evidence to indicate a significant rank correlation between the two variables? Test using \(\alpha=.05\)

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