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Q 16.

Expert-verifiedFound in: Page 718

Book edition
6th

Author(s)
Daren Starnes, Josh Tabor

Pages
837 pages

ISBN
9781319113339

Spinning heads? When a fair coin is flipped, we all know that the probability the coin lands on heads is $0.50$ However, what if a coin is spun? According to the article “Euro Coin Accused of Unfair Flipping” in the New Scientist, two Polish math professors and their students spun a Belgian euro coin $250$ times. It landed heads $140$ times. One of the professors concluded that the coin was minted asymmetrically. A representative from the Belgian mint indicated the result was just chance. Assume that the conditions for inference are met.

a. Carry out a chi-square test for goodness of fit to test if heads and tails are equally likely when a euro coin is spun.

b. In Chapter $9$ Exercise $50$ you analyzed these data with a one-sample $z$ test for a proportion. The hypotheses were ${H}_{0}:p=0.5$ and ${H}_{a}:p\ne 0.5$

where $p=$the true proportion of heads. Calculate the $z$ statistic and P-value for this test. How do these values compare to the values from part (a)?

Part (a) When a euro coin is spun, there is no persuasive evidence that heads and tails are not equally likely.

Part (b) When a euro coin is spun, there is no persuasive evidence that heads and tails are not equally likely.

Sample size $=n=25$

Level of significance $=\alpha =0.05$

Test statistic: ${\chi}^{2}=\Sigma \frac{{(O-E)}^{2}}{E}$

The null and alternative hypotheses:

${H}_{0}:p1=p2=\frac{1}{2}=0.5\phantom{\rule{0ex}{0ex}}{H}_{a}:Atleastoneofthe{p}_{i}\u2019sisincorrect.$Expected values can be found as,

${E}_{1}={np}_{1}=250\times 0.5=125\phantom{\rule{0ex}{0ex}}{E}_{2}={np}_{2}=250\times 0.5=125$Therefore, test statistic is,

${\chi}^{2}=\frac{{(140-125)}^{2}}{125}+\frac{{(110-125)}^{2}}{125}=3.6$

P-value using excel formula, $=CHIDIST(3.6,1)$

$P-$value $=0.0578$

Decision: $P-value>0.05,failtoreject{H}_{0}$

When a euro coin is spun, there is no persuasive evidence that heads and tails are not equally likely.

The null and alternative hypotheses:

${H}_{0}:p=0.5\phantom{\rule{0ex}{0ex}}{H}_{a}:p\ne 0.5$

Using excel,

Decision: $P-value>0.05,failtoreject{H}_{0}$

When a euro coin is spun, there is no persuasive evidence that heads and tails are not equally likely.

Here, ${z}^{2}=(1.90)2=3.6={\chi}^{2}$

Also, $P-$value is same for both tests.

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