SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Statistics, EXT1 S1 2025 HSC 13d

A bag contains counters, some of which are green.

One hundred trials of an experiment are run. In each trial, one counter is selected from the bag at random and its colour noted. The counter is returned to the bag after each trial.

Let \(X\) be the random variable representing the number of times that a green counter is selected.

Given that  \(E(X)=20\)  and  \(P(X \geq k)=0.0668\), find the value of \(k\). Use of a standard normal approximation table is allowed.   (4 marks)

Show Answers Only

\(k=26\)

Show Worked Solution

\(E(X)=20, n=100\)

\(E(X)=np \ \ \Rightarrow \ \ 20=100 p\ \ \Rightarrow \ \ p=\dfrac{1}{5}\)

\(X \sim B(n, p) \sim B\left(100, \dfrac{1}{5}\right)\) 

\(P(X \geqslant k)=0.0668\) 

\(1-0.0668=0.9332\)

\(\text{Using Normal Dist Table (ref = 0.9332):}\)

\(P(X \geqslant k)=P(z \geqslant 1.5)\)

\(\operatorname{Var}(X)=np(1-p)=20\left(1-\dfrac{1}{5}\right)=16\)

\(\sigma(X)=\sqrt{16}=4\)
 

\(\text{Using}\ \ z=\dfrac{x-\mu}{\sigma}:\)

\(\dfrac{k-20}{4}\) \(=1.5\)
\(k-20\) \(=6\)
\(k\) \(=26\)

Statistics, EXT1 S1 2025 HSC 4 MC

A Bernoulli random variable \(X\) has probability distribution

\(P(x)=\dfrac{x+1}{3}\)  for  \(x=0,1\).

What are the mean and variance of \(X\) ?

  1. \(E(X)=\dfrac{1}{3}, \quad \operatorname{Var}(X)=\dfrac{2}{9}\)
  2. \(E(X)=\dfrac{1}{3}, \quad \operatorname{Var}(X)=\dfrac{2}{3}\)
  3. \(E(X)=\dfrac{2}{3}, \quad \operatorname{Var}(X)=\dfrac{2}{9}\)
  4. \(E(X)=\dfrac{2}{3}, \quad \operatorname{Var}(X)=\dfrac{2}{3}\)
Show Answers Only

\(C\)

Show Worked Solution

\(P(0)=\dfrac{1}{3}, \ P(1)=\dfrac{2}{3} \)

\(E(X) = \dfrac{1}{3} \times 0 + \dfrac{2}{3} \times 1 = \dfrac{2}{3}\)

\(E(X^2) = \dfrac{1}{3} \times 0^2 + \dfrac{2}{3} \times 1^2 = \dfrac{2}{3} \)

\(\text{Var}(X) = E(X^2)-E(X)^2 = \dfrac{2}{3}-\dfrac{4}{9}=\dfrac{2}{9} \)

\(\Rightarrow C\)

Statistics, EXT1 S1 2024 MET1 4

Let \(X\) be a binomial random variable where  \(X \sim \operatorname{Bi}\left(4, \dfrac{9}{10}\right)\).

  1. Find the standard deviation of \(X\).   (1 mark)

  2. Find  \(\operatorname{Pr}(X<2)\).   (2 marks)

Show Answers Only

a.    \(\operatorname{sd}(X)=\dfrac{3}{5}\)

b.    \(\dfrac{37}{10\,000}\)

Show Worked Solution

a.     \(\operatorname{sd}(X)\) \(=\sqrt{np(1-p)}\)
    \(=\sqrt{4\times\dfrac{9}{10}\times\dfrac{1}{10}}\)
    \(=\sqrt{\dfrac{36}{100}}\)
    \(=\dfrac{3}{5}\)

 

b.     \(\operatorname{Pr}(X<2)\) \(=\operatorname{Pr}(X=0)+\operatorname{Pr}(X=1)\)
    \(=\ ^4C _0\left(\dfrac{9}{10}\right)^0\left(\dfrac{1}{10}\right)^4+\ ^4C_1\left(\dfrac{9}{10}\right)^1\left(\dfrac{1}{10}\right)^3\)
    \(= 1 \times \dfrac {1}{10\,000} + 4 \times \dfrac{9}{10} \times \dfrac{1}{1000}\)
    \(=\dfrac{37}{10\,000}\)
Mean mark (b) 51%.

Statistics, EXT1 S1 2020 HSC 12b

When a particular biased coin is tossed, the probability of obtaining a head is `3/5`.

This coin is tossed 100 times.

Let `X` be the random variable representing the number of heads obtained. This random variable will have a binomial distribution.

  1. Find the expected value, `E(X)`.  (1 mark)

  2. By finding the variance, `text(Var)(X)`, show that the standard deviation of `X` is approximately 5.  (1 mark)

  3. By using a normal approximation, find the approximate probability that `X` is between 55 and 65.  (1 mark)

Show Answers Only
  1. `60`
  2. `text(See Worked Solutions)`
  3. `68text(%)`
Show Worked Solution

i.   `X = text(number of heads)`

`X\ ~\ text(Bin) (n, p)\ ~\ text(Bin) (100, 3/5)`

`E(X)` `= np`
  `= 100 xx 3/5`
  `= 60`

 

ii.    `text(Var)(X)` `= np(1 – p)`
    `= 60 xx 2/5`
    `= 24`

 

`sigma(x)` `= sqrt24`
  `~~ 5`

 

iii.    `P(55 <= x <=65)` `~~ P(−1 <= z <= 1)`
    `~~ 68text(%)`

Statistics, EXT1 S1 EQ-Bank 10

Four cards are placed face down on a table. The cards are made up of a Jack, Queen, King and Ace.

A gambler bets that she will choose the Queen in a random pick of one of the cards.

If this process is repeated 7 times, express the gambler's success as a Bernoulli random variable and calculate

  1. the mean.  (1 mark)

  2. the variance.  (1 mark)

Show Answers Only
  1. `7/4`
  2. `21/16`
Show Worked Solution

i.     `text(Let)\ \ X = text(number of Queens chosen)`

`X\ ~\ text(Bin) (7,1/4)`

`E(X)` `=np`
  `= 7 xx 1/4`
  `=7/4`

 

ii.   `text(Var)(X)` `= np(1-p)`
    `=7/4(1-1/4)`
    `= 21/16`

Statistics, EXT1 S1 SM-Bank 4

In an experiment, a pair of dice are rolled 70 times.

A success is recorded if the sum of the dice roll is 5 or less.

  1. What is the mean of this binomial distribution?

     

    Give your answer to one decimal place.  (3 marks)

  2. What is the standard deviation?

     

    Give your answer to one decimal place.  (1 mark)

Show Answers Only
  1. `19.4`
  2. `3.7\ \ (text(to 1 d.p.))`
Show Worked Solution

i.   `text(Array of possible roll totals:)`

STRATEGY: A table (or array) can be a very efficient and error minimising strategy in questions like this.

`P(5\ text(or less)) = 10/36 = 5/18`

`text(Let)\ X =\ text(number of rolls) <= 5`

`X\ ~\ text(Bin)(70, 5/18)`

`E(X)` `= np`
  `= 70 xx 5/18`
  `= 19.4`

 

ii.    `text(Var)(X)` `= np(1 – p)`
  `sigma^2` `= 70 xx 5/18(1 – 5/18)`
    `= 14.043`
  `:. sigma` `= 3.7\ \ (text(to 1 d.p.))`

Statistics, EXT1 S1 SM-Bank 4 MC

When a standard 6-sided die is thrown, the probability that it shows a prime number is  `2/3`.

If 10 standard dice are thrown, the number, `N`, of times a prime number is showing has a binomial distribution.

What is the standard deviation of  `N`, correct to 3 decimal places?

  1. 0.222
  2. 0.471
  3. 1.491
  4. 2.222
Show Answers Only

`C`

Show Worked Solution

`N\ ~\ text(Bin)(n, p)\ ~\ text(Bin)(10, 2/3)`

`text(Var)(N)` `= np(1 – p)`
  `= 10 · 2/3(1 – 2/3)`
  `= 20/9`

 

`:. σ_N` `= sqrt(20/9)`
  `= 1.4907…`

 
`=>\ C`