Populations and Samples

NZ Level 8 (NZC) Level 3 (NCEA) [In development]

The exact distribution of the sample proportions

A dog has three puppies.

Let $M$`M` represent the number of male puppies in this litter.

a

If a dog has $3$3 puppies, then the number of male puppies, $M$`M`, can be $0$0, $1$1, $2$2 or $3$3.

What are the values of the proportions, $\hat{P}$^`P` of male puppies in the litter associated with each outcome of $M$`M`?

If $M=0$`M`=0: $\hat{P}$^`P`$=$=$\editable{}$

If $M=1$`M`=1: $\hat{P}$^`P`$=$=$\editable{}$

If $M=2$`M`=2: $\hat{P}$^`P`$=$=$\editable{}$

If $M=3$`M`=3: $\hat{P}$^`P`$=$=$\editable{}$

b

Construct the probability distribution for $M$`M` and $\hat{P}$^`P` below.

$m$m |
$0$0 | $1$1 | $2$2 | $3$3 |
---|---|---|---|---|

$P$P$($($M=m$M=m$)$) |
$\frac{1}{8}$18 | $\editable{}$ | $\editable{}$ | $\editable{}$ |

$\hat{p}$^p |
$0$0 | $\frac{1}{3}$13 | $\frac{2}{3}$23 | $1$1 |

$P$P$($($\hat{P}=\hat{p}$^P=^p$)$) |
$\editable{}$ | $\frac{3}{8}$38 | $\editable{}$ | $\editable{}$ |

c

Use your answers from part (b) to determine $P$`P`$($($\hat{P}>\frac{1}{2}$^`P`>12$)$).

Easy

Approx 5 minutes

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Make inferences from surveys and experiments: A determining estimates and confidence intervals for means, proportions, and differences, recognising the relevance of the central limit theorem B using methods such as resampling or randomisation to assess

Use statistical methods to make a formal inference