135k views
1 vote
Euro Coin. Statistics students at the Akademia Podlaka conducted an experiment to test the hypothesis that the one-Euro coin is biased (i.e., not equally likely to land heads up or tails up). Belgian-minted one-Euro coins were spun on a smooth surface, and 140 out of 250 coins landed heads up. Does this result support the claim that one-Euro coins are biased

1 Answer

6 votes

Answer:


z=\frac{0.56 -0.5}{\sqrt{(0.5(1-0.5))/(250)}}=1.897


p_v =2*P(z>1.897)=0.0578

If we compare the p value obtained and the significance level assumed
\alpha=0.05 we see that
p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the proportion of heads in the Euro coins is not significantly different from 0.5.

Explanation:

1) Data given and notation

n=250 represent the random sample taken

X=140 represent the number of heads obtained


\hat p=(140)/(250)=0.56 estimated proportion of heads


p_o=0.5 is the value that we want to test


\alpha represent the significance level

z would represent the statistic (variable of interest)


p_v represent the p value (variable of interest)

2) Concepts and formulas to use

We need to conduct a hypothesis in order to test the claim that that one-Euro coins are biased, so the correct system of hypothesis are:

Null hypothesis:
p=0.5

Alternative hypothesis:
p \\eq 0.5

When we conduct a proportion test we need to use the z statisitc, and the is given by:


z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}} (1)

The One-Sample Proportion Test is used to assess whether a population proportion
\hat p is significantly different from a hypothesized value
p_o.

Check for the assumptions that he sample must satisfy in order to apply the test

a)The random sample needs to be representative: On this case the problem no mention about it but we can assume it.

b) The sample needs to be large enough


np_o =250*0.5=125>10


n(1-p_o)=250*(1-0.5)=125>10

3) Calculate the statistic

Since we have all the info requires we can replace in formula (1) like this:


z=\frac{0.56 -0.5}{\sqrt{(0.5(1-0.5))/(250)}}=1.897

4) Statistical decision

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.

The next step would be calculate the p value for this test.

Since is a bilateral test the p value would be:


p_v =2*P(z>1.897)=0.0578

If we compare the p value obtained and the significance level assumed
\alpha=0.05 we see that
p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the proportion of heads in the Euro coins is not significantly different from 0.5.

User Duduwe
by
8.3k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories