Chapter 10.1, Problem 10.31E

(a)

To determine

To Explain: a goodness of fit test to see if there is sufficient evidence to conclude that spinning the coin does not produce an equal proportion of heads and tails.

Answer to Problem 10.31E

There is enough evidence to help the claim that spinning the coin does not produce an equal proportion of heads and tails.

Explanation of Solution

Given:

  $\begin{array}{c}n=200\\ x=122\\ n-x=200-122=78\end{array}$

Formula used:

  $\begin{array}{c}{\chi }^2={\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}}\\ E=np\end{array}$

Calculation:

Given:

  $\begin{array}{c}n=200\\ x=122\\ n-x=200-122=78\end{array}$

Assume

  $\alpha =0.05$

The null hypothesis statement is that the population proportions are equal to the given proportions

  $H_0:p_1=p_2=\frac{1}{2}=0.5$

The alternative hypothesis statement is the opposite of the null hypothesis:

  $H_1:\text{At latest one of the }{p}_i\text{'}\text{s is different}\text{.}$

The expected frequencies $E$ are

  $\begin{array}{c}{E}_1=n{p}_1=200\times 0.5=100\\ E_2=n{p}_2=200\times 0.5=100\end{array}$

The value of the test- statistic is then

  $\begin{array}{c}{\chi }^2={\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}}\\ =\frac{{\left(122-100\right)}^2}{100}+\frac{{\left(78-100\right)}^2}{100}\\ =9.68\end{array}$

The degree of freedom is

  $df=c-1=2-1=1$

The P-Value is the probability of getting the value of the test statistic, or a value more extreme and $df=1$ :

  $0.001<P<0.01$

If the P-value is equal or less than to the significance level, then the alternative hypothesis is accepted and null hypothesis is rejected:

  $P<0.5\Rightarrow {\text{Reject H}}_0$

There is enough evidence to help the claim that spinning the coin does not produce an equal proportion of heads and tails.

(b)

To determine

To Explain: whether spinning the coin is equally likely to result in heads or tails.

Answer to Problem 10.31E

There is enough evidence to help the claim that spinning the coin does not produce an equal proportion of heads and tails.

Explanation of Solution

Given:

  $p=\frac{1}{2}=0.5$

Formula used:

Calculation:

The claim is either the null hypothesis or the alternative hypothesis

  $\begin{array}{c}{H}_0:p=\frac{1}{2}=0.5\\ H_1:p\ne 0.5\end{array}$

The sample proportion is the number of outcomes divided by the sample size:

  $\widehat{p}=\frac{x}{n}=\frac{122}{200}=0.61$

Test-statistic is

  $\begin{array}{c}z=\frac{\widehat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}=\frac{0.61-0.5}{\sqrt{\frac{0.5\left(1-0.5\right)}{200}}}\\ =3.11\end{array}$

The P-Value is the probability of getting the value of the test statistic, or a value more extreme

  $P=P\left(Z<-3.11\text{}\text{or}\text{Z >3}\text{.11}\right)=2P\left(Z<-3.11\right)=2\left(0.0009\right)=0.0018$

If the P-value is smaller than the significance level $\alpha$ , then reject the null hypothesis:

  $P<0.05\Rightarrow \text{Reject }{H}_0$

There is enough evidence to help the claim that spinning the coin does not produce an equal proportion of heads and tails.

(c)

To determine

To Explain: the results of part (a) and (b) comparatively.

Answer to Problem 10.31E

Same conclusion

Explanation of Solution

It observed that the part (a) and part (b) saying the same conclusion, there is no difference in the in part (a) and part (b).