Chapter 19, Problem 19.12AYK

(a)

To determine

To explain:

The population of in that individual the parameter $p$ of the given sample.

Answer to Problem 19.12AYK

  $p\approx 0.609514$

Explanation of Solution

Given:

  $\begin{array}{l}{q}_0=0.5\\ x=615\\ N=1009\end{array}$

Concept used:

Formula

  $p=\frac{x}{n}$

Significance level

  $\begin{array}{l}\alpha =1-\text{confidence}\\ \alpha =1-0.95\end{array}$

Calculation:

  $X$ Is a random variable

  $X$ Is the number of people try to avoid drinking soda

  $\begin{array}{l}{H}_0:p=0.5\\ H_0:p>0.5\\ p_0=0.5\end{array}$

Specific value of proportion

  $\begin{array}{l}{q}_0=1-p_0\\ q_0=0.5\\ x=615\\ N=1009\end{array}$

Parameter $p=$ true proportion of individuals in population

The best point of estimate of $p$ is sample proportion

  $\begin{array}{l}p=\frac{x}{n}\\ p=\frac{615}{1009}\\ p\approx 0.609514\end{array}$

The proportion of individual in the sample try to avoid drinking soda is $0.609514$ .

(b)

To determine

To explain:The null and alternative hypothesis corresponding to Gallup’s title of the given sample.

Answer to Problem 19.12AYK

  $p\approx 0.609514$

Explanation of Solution

Given:

  $\begin{array}{l}{q}_0=0.5\\ x=615\\ N=1009\end{array}$

Concept used:

Formula

  $p=\frac{x}{n}$

Significance level

  $\begin{array}{l}\alpha =1-\text{confidence}\\ \alpha =1-0.95\end{array}$

Calculation:

Null hypothesis $\left(H_0\right)$ :-The hypothesis of no difference is called null hypothesis. Null hypothesis is the hypothesis which is tested for the possible rejection under the assumption that is true.

Alternative hypothesis $\left(H_1\right)$ :-The complementary hypothesis to the null hypothesis is called Alternative hypothesis

  $X$ Is a random variable

  $X$ Is the number of people try to avoid drinking soda

Null hypothesis $H_0:p=0.5$ [only half of the people try to avoid drinking soda]

Alternative hypothesis $H_1:p>0.5$ [majority of people try to avoid drinking soda]

Alternative hypothesis to be one sided.

Here, only values greater than $0.5$ are to be tested.

That is only one direction areto be tested.

(c)

To determine

To explain:

The test statistic and $p$ value of the given sample.

Answer to Problem 19.12AYK

  $p-$ Value $=0.000$

Explanation of Solution

Given:

  $\begin{array}{l}{q}_0=0.5\\ x=615\\ N=1009\end{array}$

Concept used:

Formula

  $p=\frac{x}{n}$

Significance level

  $\begin{array}{l}\alpha =1-\text{confidence}\\ \alpha =1-0.95\end{array}$

Calculation:

  $X$ Is a random variable

  $X$ Is the number of people try to avoid drinking soda

  $\begin{array}{l}{H}_0:p=0.5\\ H_0:p>0.5\\ p_0=0.5\end{array}$

Specific value of proportion

  $\begin{array}{l}{q}_0=1-p_0\\ q_0=0.5\\ x=615\\ N=1009\end{array}$

Parameter $p=$ true proportion of individuals in population

The best point of estimate of $p$ is sample proportion

  $\begin{array}{l}p=\frac{x}{n}\\ p=\frac{615}{1009}\\ p\approx 0.609514\end{array}$

Test statistic

The value of $p$ is

  $\hat{p}\sim N\left(p,\frac{pq}{n}\right)$

  $p$ Comes from null hypothesis

  $\begin{array}{l}z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}\\ z_0=6.96\end{array}$

  $p-$ Value

  $\begin{array}{l}=P\left(z>z_0\right)\\ z\sim N\left(0,1\right)\end{array}$

  $p-$ Value $=0.000$

A $p-$ Value is nearly zero

So, the null hypothesis are rejected.