Chapter 9.2, Problem 9.21E

a)

To determine

To find the mean and standard deviation of the proportion $\widehat{p}$ .

Answer to Problem 9.21E

Mean = 0.5 and standard deviation = 0.1

Explanation of Solution

Given:

Sample size = n = 25

Formula:

Mean:

  ${\mu }_{\widehat{p}}=p$

Standard deviation:

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

Calculation:

A standard deck of cards contains 52 cards, of which 26 are red and 26 are black, 13 are of each suit (hearts, diamonds, spades, clubs) and of which 4 are of each denomination (A, 2 to 10, J. Q, K). The face cards are the jacks J. queens Q and kings K. Therefore, 26 of the 52 cards in the deck are red.

So, the probability is the number of favorable outcomes divided by the number of possible outcomes

  $=\frac{26}{52}=0.5$

Sample proportion = $\widehat{p}$ = 0.5

The mean = ${\mu }_{\widehat{p}}=p=0.5$

The standard deviation:

  ${\sigma }_{\widehat{p}}=\sqrt{\frac{0.5(1-0.5)}{25}}\approx 0.1$

b)

To determine

To find proportion $\widehat{p}$ if red cards are 16.

Answer to Problem 9.21E

Sample proportion = 0.64

Explanation of Solution

Given:

  $x=16$

Formula:

  $\widehat{p}=\frac{x}{n}$

Calculation:

Using the formula,

  $\widehat{p}=\frac{16}{25}=0.64$

c)

To determine

To sketch the graph for part b)

Explanation of Solution

Given:

  $\widehat{p}=0.64$

Calculation:

Following is the graph which shows sampling distribution of sample proportion = 0.64:

d)

To determine

To explain how likely is it that you would get a value of 19 as unusual as the one you did if the deck is fair.

Explanation of Solution

Given:

  $\widehat{p}=0.64$

Population proportion = p = 0.5

Formula:

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

Calculation:

First need to find z-score:

  $z=\frac{0.64-0.5}{\sqrt{\frac{0.5(1-0.5)}{25}}}=1.40$

Therefore,

  $\begin{array}{l}P(\widehat{p}>0.64)=P(Z>1.40)\\ P(\widehat{p}>0.64)=1-P(Z<1.40)\\ P(\widehat{p}>0.64)=1-0.9192=0.0808\end{array}$

Using excel formula, =NORMSDIST(1.4)

Hence, the sample proportion of 0.64 is likely to occur by chance as probability is more than 0.05. Thus, deck does not indeed appear to be fair.