Chapter 9.2, Problem 50E

a.

To determine

The appropriate hypotheses for the true proportions of spins are to be explained.

Answer to Problem 50E

The appropriate hypotheses for the true proportions of spins are:

The null hypothesis: $H_0:\text{}p=0.50$

The alternate hypothesis: $H_a:\text{}p\ne 0.50$

Explanation of Solution

Given:

  $\text{}p=0.50$

On flipping of the fair coin, there is probability of coin lands on heads is $0.50$

So, the null hypothesis is given by the following expression:

  $H_0:\text{}p=0.5$

Exactly opposite to the null hypothesis is stated by the alternate hypothesis:

  $H_a$ : at least $p$ is incorrect.

Or

  $H_a:\text{}p\ne 0.50$

Conclusion:

Hence, null hypothesis and alternate hypothesis are stated above.

b.

To determine

The $P-$ value and standardized test statistics are to be estimated.

Answer to Problem 50E

The $P-$ value and standardized test statistics are $0.11507,1.2$ respectively.

Explanation of Solution

Given:

  $\begin{array}{l}n=250\\ x=140\\ p=0.50\end{array}$

Concept Used:

The test statistic is calculated by using the following expression:

  $Z=\frac{\widehat{p}-p}{\sqrt{\frac{p\left(1-p\right)}{n}}}$

Where,

Sample proportion: $\widehat{p}$

Sample size: $n$

Calculation:

Firstly need to calculate the sample proportion which is the ratio of the number of successes and the sample size:

  $\begin{array}{l}\widehat{p}=\frac{x}{n}\\ \widehat{p}=\frac{140}{250}\\ \widehat{p}=0.56\end{array}$

The test statistic is calculated by using the following expression:

  $Z=\frac{\widehat{p}-p}{\sqrt{\frac{p\left(1-p\right)}{n}}}$

Put the values from given data in the above eq.

  $\begin{array}{l}Z=\frac{\widehat{p}-p}{\sqrt{\frac{p\left(1-p\right)}{n}}}\\ Z=\frac{0.56-0.5}{\sqrt{\frac{0.5\left(1-0.5\right)}{100}}}\\ Z=\frac{0.56-0.5}{\sqrt{\frac{0.5\left(0.5\right)}{100}}}\\ Z=\frac{0.06}{0.05}\\ Z=1.2\end{array}$

Now the probability value of the $Z-$ score $Z=1.2$ is:

  $\begin{array}{l}P\left(x<Z\right)=0.88493\\ P\left(x>Z\right)=1-0.88493=0.11507\end{array}$

Conclusion:

By using the chi-square subtotals, value of the test-statistics is estimated.

c.

To determine

The meaning of the $P-$ value and conclusion of the claim of competing for true proportions is to be interpreted.

Answer to Problem 50E

  $P-$ value is less than the significance level that means reject hypothesis.

Explanation of Solution

If the $P-$ value is less than or equal to the significance level then the null hypothesis is rejected.

If the $P-$ value is greater than the significance level that means fail to reject null hypothesis.

From the answer of above part:

  $p=0.11507$

So, $P$ value is $0.11507$ which less than the given significance level is of $0.50$ . So, it leads to reject the null hypothesis $H_0:p=0.50$ .

So, it clearly shows that it has enough evidence to claim the competing about the true proportions of spin.

Conclusion:

Hence, the null hypothesis $H_0$ is rejected.