Explor 1.5 Angie Ashby
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Hollins University *
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Apr 3, 2024
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E
XPLORATION 1.5:
Eye Dominance
pg. 1
Exploration 1.5: Eye Dominance Name(s): Angie Ashby Just like handedness where people prefer to use one hand over another, eye dominance, sometimes called eyedness, is the tendency to prefer to see using one eye over the other. Interestingly, the side of the dominant eye does not always match that of the dominant hand. Let’s investigate whether people are equally likely to have left-eye or right-eye dominance by collecting some data from you and your classmates. 1.
To figure out which of your eyes is the dominant eye, carry out the following “dominant eye test”: •
Extend both your arms in front of you and create a triangular opening using your thumbs and pointer fingers. •
With both eyes open, center your triangular opening on a distant object such as a clock or projector. •
Close your left eye. If the object stays centered in your triangular opening, your right eye (as that is the one that’s open) is your dominant eye. If the object is no longer in the triangular opening your left eye is your dominant eye. •
Double check this by closing your right eye. If the object stays centered in your triangular opening, your left eye (as that is the one that’s open) is your dominant eye. If the object is no longer in the triangular opening, your right eye is your dominant eye. Record whether you have left-eye or right-eye dominance. right-eye
Check the course schedule page to find the results from the entire class. Before we combine your data with the data from your classmates, let’s think about what we want to test here. Conventional wisdom says that more often people are right-handed than left, so for now let’s use our research hypothesis to be that more often people tend to have right-eye dominance than left-eye dominance. 2.
In this study with your classmates: (a)
What are the observational units in this study? [ The students whose eyes are being tested ] (b)
What is the variable that is recorded? [ Left or right eye dominance ] (c)
Describe the parameter of interest in words. (Use the symbol π to represent this parameter.) [ π= probability of people being more
right eye dominant ]
(d)
If right-eye and left-eye dominance are equally prevalent, what would you expect the numerical value of the parameter to be? Is this the null hypothesis or the alternative hypothesis? [ Null = .5 ]
(e)
If people are more likely to be right-eye dominant that left, what can you say about the numerical value of the parameter? Is this the null hypothesis or the alternative hypothesis? [ Alternative = > .5 ]
E
XPLORATION 1.5:
Eye Dominance
pg. 2
3.
These data were collected in the Data Collection survey on the first day of class. The results are posted on the course schedule page. There were 36 total responses
with 12 left-eye dominant
and 24 right-eye dominant
. Calculate and report the sample proportion who are right-eye dominant. [ 24/36 sample proportion = .67 ] 4.
To have a larger sample size to analyze, combine
your class results with the results from some of the author’s classes, in which 70 of 115
students had right-eye dominance. Now what are the sample size and the sample proportion that are right-eye dominant? Sample size: [ 94/151 ] Sample proportion: [ .62 ] 5.
Use the One Proportion
applet to test the hypotheses from #2d and #2e. (a)
Describe the shape of the null distribution of sample proportions. Does this shape look familiar? Where is the null distribution centered? Does this make sense? Check the Summary stats
box and report the mean and standard deviation as reported by the applet. Shape: [ normal, bell shaped] Familiar? [ Yes ] Center?
[ 0.5 ] Why does this make sense? [ The null must be true? ] Mean:
[0.501 ] SD: [ 0.041 ] (b)
Approximate the p-value and summarize the strength of evidence that the sample data provide regarding the research hypothesis. [ 0.004 , there is very strong evidence against the null ] (c)
Determine the standardized statistic, z
, and summarize the strength of evidence. Confirm that the strength of evidence obtained using the standardized statistic is similar to that obtained using the p-value. [ (0.62 - .501) / .041= 2.9 ] Theory-based Approach (One-Proportion z
-test) In #5a, you probably described the shape of the null distribution using words such as bell-shaped, symmetric, or maybe even normal. You have seen many null distributions in this chapter that have had this same basic shape. You should have also noticed that the null distributions have all been centered at the hypothesized value of the long-run proportion used in the null hypothesis. You probably could have predicted that your null distribution was going to be somewhat bell-shaped and centered at 0.50. You probably would have a harder time predicting your null distribution’s variability (standard deviation), but this too can be predicted in advance, as we will see shortly. We can use mathematical models known as normal distributions (bell-shaped curves) to approximate many of the null distributions we have generated so far in this text. When rules and theories are used to predict what the value of the standardized statistic and p-value would be if someone carried out a simulation, we call the approach a theory-based approach
. The normal distribution provides a second way, in addition to simulation, to approximate a p-value.
E
XPLORATION 1.5:
Eye Dominance
pg. 3
6.
Check the box next to Normal Approximation in the applet. Does the region shaded in blue seem to be a good description (model) of what we actually got in the simulation? Yes, the normal approximation is very accurate to the null distribution Validity Conditions for Theory-Based Approach The normal approximation to the null distribution is valid whenever the sample data is reasonably large. One convention is to consider the sample size large enough whenever there are at least 10 observations in each category. Validity Conditions The normal approximation can be thought of as a prediction of what would occur if a simulation-
based analysis was carried out. Many times, this prediction is valid, but not always. It is only valid when the condition (at least 10 successes and at least 10 failures) is met. 7.
According to this convention, is the sample size large enough in this study to use the normal approximation and theory-based inference? Justify your answer. [ Yes, we performed the observation 151 times and got 70 successes (right eye) and 81 failures (left eye)] Formulas The normal approximation will also give you values of the standardized statistic and p-value based on its mathematical predictions. As you learned in Section 1.3, the standardized score is calculated as The mean of the null distribution is the hypothesized value of the long-run proportion (π). The standard deviation can be obtained in two ways: First, find the standard deviation of the null distribution by simulating. Second, predict the value of the standard deviation by substituting into this formula: 8.
Use the formula to determine the (theoretical/ predicted) standard deviation of the sample proportion. Then compare this to the SD from your simulated sample proportions as recorded in #5a. Are they similar? [ 0.5(1-0.5)/151= 0.041, Yes they are similar the SD I got was 0.041 ] The predicted value of the standard deviation (using the formula) will be very close to the simulated standard deviation of the null distribution. The validity condition mentioned earlier says the shape will be approximately normal when the sample size is large enough where, a “large enough” sample size means at least 10 successes and at least 10 failures. This mathematical prediction is often called the “central limit theorem. Central Limit Theorem .
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=
statistic
mean of null distribution
z
standard deviation of null distribution
(1
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E
XPLORATION 1.5:
Eye Dominance
pg. 4
If the sample size (
n
) is large enough, the distribution of sample proportions will be bell-shaped (or normal), centered at the long-run proportion (π), with a standard deviation of
9.
Use the predicted value of the standard deviation from #8 to calculate the standardized statistic (
z
) by hand and confirm that your answer is very close to what you found in #5c when using simulation. [ 0.62- 0.5 /0.041= 2.9, My answer was exactly right! ] 10.
The theory-based (normal approximation) p-value is also now displayed. Compare this p-value to the one you got from simulation (#5b). Are they similar? [ Simulated 0.0040, Theory-based 0.0073 , they’re within range of each other ] 11.
Why are the standard deviation (#8), standardized statistic (#9) and p-value (#10) similar when using the theory-based (one proportion z
-test / normal approximation) to what you got in your simulation? When would they be different? [ They are similar because we are expecting the null to be true , and when we perform larger samples we get closer to the true probability(normal approximation) ] Exploring Further Follow-up Analysis #1 There are several research papers (see, for example, Ehrenstein et al., 2005) on eye preference that say that the long-run proportion of right-eyed people is two-thirds.
12.
Use the theory-based approach to test the claim that there is a two-thirds chance (π = 0.667) that a person will be right-eye dominant, using the sample data (your classmates combined with an author’s class) and a two-sided alternative
. Report the null and alternative hypotheses, standardized statistic, and p-value. Summarize your conclusion and explain the reasoning process by which it follows from your analysis. [ Null π = 2/3, Alternative π ≠ 2/3 , z = -1.4, p-value= 0.92 , There would be little evidence against the null because of the small z and large p-value. Meaning being 2/3 right eye dominant is probably true ] Follow-up Analysis #2 In a small class of 14 students, 9 students turned out to be right-eye dominant. 13.
Use simulation to generate a two-sided p-value evaluating the strength of evidence that the long-run proportion of right-eyed students is different from 50% based on this small class’s data alone. [ 0.416 ] 14.
Why can’t you use the normal approximation in this case? [ it has to add the tails on both sides ] 15.
Use the normal approximation anyway. Compare and comment on the p-values obtained from the two methods. [ normal p-value 0.29, it is double what the normal approximation p-value is, since its two-sided, theory based would be ineffective. ] (1
) /
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