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New Zealand
Level 8 - NCEA Level 3

The exact distribution of the sample proportions

Interactive practice questions

A dog has three puppies.

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


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$=$=$\editable{}$


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 $\editable{}$ $\editable{}$

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

Approx 5 minutes
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Three marbles are randomly drawn from a bag containing five black and six grey marbles.

Let $X$X be the number of black marbles drawn, with replacement.

A company wants to know the likelihood of securing sales with potential clients.

Historically, the company has $70$70 successful sales for every $100$100 potential clients contacted.

Let $X$X be the number of sales the company secures within the next $4$4 potential clients.

A pencil case contains $9$9 red pens and $7$7 black pens. $4$4 pens are drawn randomly from the pencil case, one at a time, each being replaced before the next one is drawn.

Let $W$W be the number of red pens drawn.



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

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