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Sagot :
Testing the hypothesis, it is found that the p-value of the test is 0.017 < 0.05, which means that we can conclude that the red shirt increases his chances of winning.
At the null hypothesis, we test if the proportion is of 40%, that is:
[tex]H_0: p = 0.4[/tex]
At the alternative hypothesis, we test if the proportion is greater than 40%, that is:
[tex]H_1: p > 0.4[/tex]
The test statistic is:
[tex]Z = \frac{X - p}{s}[/tex]
In which:
- X is the sample proportion.
- p is the value tested at the null hypothesis.
- s is the standard error, given by:
[tex]s = \sqrt{\frac{p(1 - p)}{n}}[/tex]
With n as the sample size.
In this problem:
- 0.4 is tested at the null hypothesis, thus [tex]p = 0.4[/tex].
- Won 3 out of 3, thus [tex]n = 3, X = \frac{3}{3} = 1[/tex].
- The standard error is:
[tex]s = \sqrt{\frac{0.4(0.6)}{3}} = 0.283[/tex]
The value of the test statistic is:
[tex]Z = \frac{X - p}{s}[/tex]
[tex]Z = \frac{1 - 0.4}{0.283}[/tex]
[tex]Z = 2.12[/tex]
The p-value is the probability of finding a sample proportion of 1 or "above", which is 1 subtracted by the p-value of Z = 2.12.
- Looking at the z-table, Z = 2.12 has a p-value of 0.983.
- 1 - 0.983 = 0.017.
The p-value of the test is 0.017 < 0.05, which means that we can conclude that the red shirt increases his chances of winning.
A similar problem is given at https://brainly.com/question/24166849
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