Chapter 9.1, Problem 9.17E

a)

To determine

To explain the shape, center and spread of the sampling distribution of the sample proportion.

Answer to Problem 9.17E

${\mu }_{\widehat{p}}=0.14$ and ${\sigma }_{\widehat{p}}=0.0142$

Explanation of Solution

Given:

P = 14% = 0.14

n = 600

Formula:

Mean:

  ${\mu }_{\widehat{p}}=p$

Standard deviation:

  ${\sigma }_{\widehat{p}}=\sqrt{\frac{p(1-p)}{n}}$

Calculation:

The mean = ${\mu }_{\widehat{p}}=p=0.14$

The standard deviation:

  ${\sigma }_{\widehat{p}}=\sqrt{\frac{0.14(1-0.14)}{600}}=0.0142$

The sampling distribution of sample proportion $\widehat{p}$ is approximately normal with mean ${\mu }_{\widehat{p}}$ and ${\sigma }_{\widehat{p}}$ as population size is large (n=600)

b)

To determine

To explain whether your sample to contain 18.2% or more who own Harleys.

Explanation of Solution

Given:

Sample proportion = $\widehat{p}=0.183$

Formula:

  $z=\frac{\widehat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

Calculation:

First we need to find z-score,

  $z=\frac{0.182-0.14}{\sqrt{\frac{0.14(1-0.14)}{600}}}=2.96$

Therefore,

P(Z> 2.96) = 1-P(Z<2.96) = 1-0.9985 = 0.0015 …Using excel formula, =NORMSDIST(2.96)

This probability is less than 0.05, which indicate that the event unlikely to occur by chance.

c)

To determine

To explain whether your sample to contain less than 12% or more who own Harleys.

Explanation of Solution

Given:

Sample proportion = $\widehat{p}=0.12$

Formula:

  $z=\frac{\widehat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

Calculation:

First we need to find z-score,

  $z=\frac{0.12-0.14}{\sqrt{\frac{0.14(1-0.14)}{600}}}=-1.41$

Therefore,

P(Z< -1.41) = 0.0793 …Using excel formula, =NORMSDIST(-1.41)

This probability is more than 0.05, which indicate that the event likely to occur by chance.