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Q.2

Expert-verifiedFound in: Page 594

Book edition
4th

Author(s)
David Moore,Daren Starnes,Dan Yates

Pages
809 pages

ISBN
9781319113339

Eye black Athletes performing in bright sunlight often smear black grease under their eyes to reduce glare. Does cye black work? In one experiment, 16 randomly selected student subjects took a test of sensitivity to contrast after 3 hours facing into bright sun, both with and without eye black. Here are the differences in sensitivity, with eye black mines without eye black:

0.07 | 0.64 | -0.12 | -0.05 | -0.18 | 0.14 | -0.16 | 0.03 |

0.05 | 0.02 | 0.43 | 0.24 | -0.11 | 0.28 | 0.05 | 0.29 |

We want to know whether cye black increases sensitivity an the average.

(a) State hypotheses, Be sure to define the parameter.

(b) Check conditions for carrying out a significance test.

(c) The$Pvalue$ of the test is $0.047$. Interpet this value in context.

- Interpret a Type l error and a Type ll error in context, and give the consequences of each.

- Understand the relationsonship between the significance level at a test P(Type li error), and power.

(a)${H}_{0}:\mu d=0$

${H}_{\alpha}:\mu d>0$

(b)Conditions for carrying out a significance test is satisfied

(c) It means that at an average $4.7\%$ of athletes differ from the athletes whose eye black does not increase the sensitivity.

Given in the question that

A significance test is performed to test the sensitivity with eye black and without eye black in athletes.

Sample size $n=16$. we have to State hypotheses, Be sure to define the parameter.

The differences in sensitivity with eye black minus without eye black are shown below:

0.07 | 0.64 | -0.12 | -0.05 | -0.18 | 0.14 | -0.16 | 0.03 |

0.05 | 0.02 | 0.43 | 0.24 | -0.11 | 0.28 | 0.05 | 0.29 |

Let$\mu d$ denote the average sensitivity differential between with and without eye black.

The average sensitivity of eye black is the parameter of interest. The test is carried out to see if the average sensitivity of eye black has increased.

${H}_{0}:\mu d=0$

${H}_{\alpha}:\mu d>0$

we have to Check conditions for carrying out a significance test.

The following are the requirements for conducting a paired test: 1. A random sample must be chosen.

2. No outliers should exist.

3. The observations are unrelated to one another, or the sample is no more than $10\%$ of the total population.

The students in the sample were chosen at random.

There are no outliers in the data, as seen in the histogram below:

we know that there will be at least $160$students who are athletes.so the $10\%$ condition is also satisfied.

Given in the question that $Pvalue=0.047$ we have to Interpret this value in context.

The $P$ value of the test is $0.047$ It means that at an average $4.7\%$ of athletes differ from the athletes whose eye black does not increase the sensitivity.

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