Chapter 10.4, Problem 42E

Solution

To calculate: How many frogs in this sample must exhibit the skin discoloration in order in provide statistically significant evidence.

At least 20 frogs.

Given information:

The total sample size is 150.

Formula used:

The z-score is the value decreased by the mean np and divided by the standard deviation:

  $\sqrt{npq}=\sqrt{np\left(1-p\right)}$

  $z=\frac{x-np}{\sqrt{np\left(1-p\right)}}$

Calculation:

n =Sample size=150

p=Probability of success= $\frac{1}{12}$

The z-score is the value decreased by the mean np and divided by the standard deviation

  $\begin{array}{c}z=\frac{x-np}{\sqrt{np\left(1-p\right)}}=\frac{x-150\left(1/12\right)}{\sqrt{150\left(1/12\right)\left(1-1/12\right)}}\\ =\frac{x-12.5}{\sqrt{150\left(1/12\right)\left(1-1/12\right)}}\end{array}$

A z-score is considered unusual when it is smaller than -2 or larger than 2.

So, first convinced that ESP actually exists, when the z-score is larger than 2.

  $\frac{x-12.5}{\sqrt{150\left(1/12\right)\left(1-1/12\right)}}> 2$

Multiply each side of the previous inequality by $\sqrt{150\left(1/12\right)\left(1-1/12\right)}$ .

  $x-12.5$ >2 $\sqrt{150\left(1/12\right)\left(1-1/12\right)}$

Add 12.5 to each side of the previous inequality:

  $x$ >12.5+2 $\sqrt{150\left(1/12\right)\left(1-1/12\right)}$

Evaluate:

X>19.270032

The number of successes x needs to be an integer.

  $x\ge 20$

Thus, we would require that at least 20 frogs exhibit the skin condition in order to provide statistically significant evidence.