How do eye and head movements relate to body movements when a person reacts to a

Question

How do eye and head movements relate to body movements when a person reacts to a visual stimulus? A scientist at the California Institute of Technology designed an experiment to answer this question and reported their results in Nature. Adult male rhesus monkeys were exposed to a visual stimulus (i.e., a panel of light-emitting diodes), and their eye. head and body movements were electronically recorded. in one variation of the experiment, two variables were measured: active head movement (x. percent per degree) and body-plus-head rotation (y. percent degree). the data for n = 39 trials were subjected to a simple linear regression analysis, with the following results: beta_1 = 0.88 and use(beta_1) = 0.14. Conduct a test to determine whether the two variables, active head movement t, and the body-plus-head rotation y are positively linearly related. Use alpha = 0.05. Construct and interpret a 90% confidence interval for beta_1. the scientists want to know whether the true slope of the line differs significantly from on the basis of your answer to part (b). make the appropriate inference.

Explanation / Answer

Part a

Here, we have to use the t test for checking the slope coefficient. The null and alternative hypothesis for this test is given as below:

Null hypothesis: H0: The two variables active head movement x and the body plus head rotation y are not linearly related.

Alternative hypothesis: Ha: The two variables active head movement x and the body plus head rotation y are positively linearly related.

Symbolically, the null and alternative hypotheses are given as below:

H0: 1 = 0 versus Ha: 1 > 0

The level of significance or alpha value for this test is given as 0.05.

= 0.05

The test statistic formula is given as below:

Test statistic = t = 1-hat / se(1-hat)

We are given 1-hat = 0.88 and se(1-hat) = 0.14

Test statistic = t = 0.88 / 0.14 = 6.285714

P-value = 0.00 approximately

P-value < Alpha value

So, we reject the null hypothesis H0

We conclude that there is sufficient evidence that the two variables active head movement x and the body plus head rotation y are positively linearly related.

Part b

Confidence interval = 1-hat -/+ t* se(1-hat)

We are given confidence level = 90%

Sample size = n = 39

Degrees of freedom = n – 1 = 39 – 1 = 38

So, critical t value for df = 38 and c = 0.90 is given as below:

t = 1.6860

Lower limit = 0.88 – 1.6860*0.14 = 0.64396

Upper limit = 0.88 + 1.6860*0.14 = 1.11604

Confidence interval = (0.64, 1.12)

Part c

The value of true slope = 1 lies within the above Confidence interval = (0.64, 1.12), so we conclude that the true slope of the line does not differs significantly from 1.

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