Chapter 21, Problem 10E

(a)

To determine

To estimate the true proportion of heads and use a $95\%$ confidence interval and check the condition.

Answer to Problem 10E

  $\left(0.498,0.622\right)$ .

Explanation of Solution

It is given in the question that soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin $250$ times and gotten heads $140$ times. Thus, we have,

Let us first check the conditions for the following as:

Random condition: It is satisfied as it is assumed that the sample is randomly selected.

Normal condition: It is satisfied as,

  $\begin{array}{l}np=140>10\\ n(1-p)=110>10\end{array}$

$10\%$ condition: It is satisfied as the sample size is less than $10\%$ of the population size.

Thus, all the conditions are met. Therefore, now, to find the confidence interval we will use the calculator $\text{TI}-89$ . So, we have to first go to the $\text{Ints}$ menu and select $5:1-\text{PropZInt}$ . Then we have to enter the number of successes observed and the sample size. Then specify a confidence level. Then the interval will be calculated. Thus, we have,

  $\begin{array}{l}x=140\\ n=250\\ c=0.95\end{array}$

Thus, the confidence interval will be as:

  $\left(0.498,0.622\right)$

(b)

To determine

To explain does your confidence interval provide evidence that the coin is unfair when spun.

Answer to Problem 10E

Yes, it provide evidence.

Explanation of Solution

It is given in the question that soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin $250$ times and gotten heads $140$ times. Thus, the confidence interval will be as:

  $\left(0.498,0.622\right)$

Thus, our confidence interval provide evidence that the coin is unfair when spun because $0.50$ is within the confidence interval so it is a plausible value for the proportion of heads.

(c)

To determine

To find out what is the level of significance of this test.

Answer to Problem 10E

  $\alpha =0.05$ .

Explanation of Solution

It is given in the question that soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin $250$ times and gotten heads $140$ times. Thus, the confidence interval will be as:

  $\left(0.498,0.622\right)$

Thus, this is a two tailed test based on a $95\%$ confidence interval thus, then $\alpha =0.05$ .