Tim claims that a coin is coming up tails less than half of the time. In 110 tosses, the coin comes up tails 47 times. Using P-value, test Tim's claim. Use a 0.10 significance level and determine conclusion.

If we compare the p value obtained and using the significance level given [tex]\alpha=0.1[/tex] we have that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 10% of significance the proportion of tails is significantly less than 0.5.

Step-by-step explanation:

1) Data given and notation

n=110 represent the random sample taken

X=47 represent the number of tails obtained

[tex]\hat p=\frac{47}{110}=0.427[/tex] estimated proportion of tails

[tex]p_o=0.5[/tex] is the value that we want to test

[tex]\alpha=0.1[/tex] represent the significance level

Confidence=90% or 0.90

z would represent the statistic (variable of interest)

[tex]p_v[/tex] 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 the proportion of tails is lower than 0.5:

Null hypothesis:[tex]p\geq 0.5[/tex]

Alternative hypothesis:[tex]p < 0.5[/tex]

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

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

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

3) Calculate the statistic

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

[tex]z=\frac{0.427 -0.5}{\sqrt{\frac{0.5(1-0.5)}{110}}}=-1.531[/tex]

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 significance level provided [tex]\alpha=0.1[/tex]. The next step would be calculate the p value for this test.

Since is a left tailed test the p value would be:

[tex]p_v =P(z<-1.531)=0.0629[/tex]

If we compare the p value obtained and using the significance level given [tex]\alpha=0.1[/tex] we have that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 10% of significance the proportion of tails is significantly less than 0.5.