Questions S2 - A-Level Maths

Use an exact binomial distribution to test the claim at the $10 \%$ significance level.
A statistics teacher says that considering only the last 8 Head Students may not be satisfactory. Explain what needs to be assumed about the data for the test to be valid.

OCR S2 2016 June Q6

12 marks Moderate -0.8

6 The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by $X$. Cars pass the point at random intervals and the expected value of $X$ is denoted by $\lambda$.

State, in the context of the question, two conditions needed for $X$ to be well modelled by a Poisson distribution.
At a quiet time of the day, $\lambda = 6.50$. Assuming that a Poisson distribution is valid, calculate $\mathrm { P } ( 4 \leqslant X < 8 )$.
At a busy time of the day, $\lambda = 30$.
(a) Assuming that a Poisson distribution is valid, use a suitable approximation to find $\mathrm { P } ( X > 35 )$. Justify your approximation.
(b) Give a reason why a Poisson distribution might not be valid in this context when $\lambda = 30$.

OCR S2 2016 June Q7

11 marks Standard +0.3

7 A continuous random variable $X$ has probability density function $$f ( x ) = \left\{ \begin{array} { c c } a x ^ { - 3 } + b x ^ { - 4 } & x \geqslant 1 \\ 0 & \text { otherwise } \end{array} \right.$$ where $a$ and $b$ are constants.

Explain what the letter $x$ represents. It is given that $\mathrm { P } ( X > 2 ) = \frac { 3 } { 16 }$.
Show that $a = 1$, and find the value of $b$.
Find $\mathrm { E } ( X )$.

OCR S2 2016 June Q8

13 marks Standard +0.3

8 It is known that the lifetime of a certain species of animal in the wild has mean 13.3 years. A zoologist reads a study of 50 randomly chosen animals of this species that have been kept in zoos. According to the study, for these 50 animals the sample mean lifetime is 12.48 years and the population variance is 12.25 years $^ { 2 }$.

Test at the $5 \%$ significance level whether these results provide evidence that animals of this species that have been kept in zoos have a shorter expected lifetime than those in the wild.
Subsequently the zoologist discovered that there had been a mistake in the study. The quoted variance of 12.25 years ${ } ^ { 2 }$ was in fact the sample variance. Determine whether this makes a difference to the conclusion of the test.
Explain whether the Central Limit Theorem is needed in these tests.

OCR S2 2016 June Q9

6 marks Challenging +1.2

9 The random variable $R$ has the distribution $\operatorname { Po } ( \lambda )$. A significance test is carried out at the $1 \%$ level of the null hypothesis $\mathrm { H } _ { 0 } : \lambda = 11$ against $\mathrm { H } _ { 1 } : \lambda > 11$, based on a single observation of $R$. Given that in fact the value of $\lambda$ is 14 , find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. \section*{END OF QUESTION PAPER}

OCR S2 2009 January Q1

4 marks Moderate -0.8

1 A newspaper article consists of 800 words. For each word, the probability that it is misprinted is 0.005 , independently of all other words. Use a suitable approximation to find the probability that the total number of misprinted words in the article is no more than 6 . Give a reason to justify your approximation.

OCR S2 2009 January Q2

4 marks Standard +0.3

2 The continuous random variable $Y$ has the distribution $\mathrm { N } \left( 23.0,5.0 ^ { 2 } \right)$. The mean of $n$ observations of $Y$ is denoted by $\bar { Y }$. It is given that $\mathrm { P } ( \bar { Y } > 23.625 ) = 0.0228$. Find the value of $n$.

OCR S2 2009 January Q3

8 marks Moderate -0.3

3 The number of incidents of radio interference per hour experienced by a certain listener is modelled by a random variable with distribution $\operatorname { Po } ( 0.42 )$.

Find the probability that the number of incidents of interference in one randomly chosen hour is
(a) 0 ,
(b) exactly 1 .
Find the probability that the number of incidents in a randomly chosen 5-hour period is greater than 3.
One hundred hours of listening are monitored and the numbers of 1 -hour periods in which 0,1 , $2 , \ldots$ incidents of interference are experienced are noted. A bar chart is drawn to represent the results. Without any further calculations, sketch the shape that you would expect for the bar chart. (There is no need to use an exact numerical scale on the frequency axis.)

OCR S2 2009 January Q4

10 marks Moderate -0.3

4 A television company believes that the proportion of adults who watched a certain programme is 0.14 . Out of a random sample of 22 adults, it is found that 2 watched the programme.

Carry out a significance test, at the $10 \%$ level, to determine, on the basis of this sample, whether the television company is overestimating the proportion of adults who watched the programme.
The sample was selected randomly. State what properties of this method of sampling are needed to justify the use of the distribution used in your test.

OCR S2 2009 January Q5

9 marks Standard +0.3

5 The continuous random variables $S$ and $T$ have probability density functions as follows. $$\begin{array} { l l } S : & \mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases} \\ T : & \mathrm { g } ( x ) = \begin{cases} \frac { 5 } { 64 } x ^ { 4 } & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases} \end{array}$$

Sketch, on the same axes, the graphs of f and g .
Describe in everyday terms the difference between the distributions of the random variables $S$ and $T$. (Answers that comment only on the shapes of the graphs will receive no credit.)
Calculate the variance of $T$.

OCR S2 2009 January Q6

11 marks Standard +0.3

6 The weight of a plastic box manufactured by a company is $W$ grams, where $W \sim \mathrm {~N} ( \mu , 20.25 )$. A significance test of the null hypothesis $\mathrm { H } _ { 0 } : \mu = 50.0$, against the alternative hypothesis $\mathrm { H } _ { 1 } : \mu \neq 50.0$, is carried out at the $5 \%$ significance level, based on a sample of size $n$.

Given that $n = 81$,
(a) find the critical region for the test, in terms of the sample mean $\bar { W }$,
(b) find the probability that the test results in a Type II error when $\mu = 50.2$.
State how the probability of this Type II error would change if $n$ were greater than 81 .

OCR S2 2009 January Q7

12 marks Standard +0.3

7 A motorist records the time taken, $T$ minutes, to drive a particular stretch of road on each of 64 occasions. Her results are summarised by $$\Sigma t = 876.8 , \quad \Sigma t ^ { 2 } = 12657.28$$

Test, at the $5 \%$ significance level, whether the mean time for the motorist to drive the stretch of road is greater than 13.1 minutes.
Explain whether it is necessary to use the Central Limit Theorem in your test.

OCR S2 2009 January Q8

14 marks Moderate -0.3

8 A sales office employs 21 representatives. Each day, for each representative, the probability that he or she achieves a sale is 0.7 , independently of other representatives. The total number of representatives who achieve a sale on any one day is denoted by $K$.

Using a suitable approximation (which should be justified), find $\mathrm { P } ( K \geqslant 16 )$.
Using a suitable approximation (which should be justified), find the probability that the mean of 36 observations of $K$ is less than or equal to 14.0 . 4

OCR S2 2011 January Q1

4 marks Easy -1.2

1 A random sample of nine observations of a random variable is obtained. The results are summarised as $$\Sigma x = 468 , \quad \Sigma x ^ { 2 } = 24820 .$$ Calculate unbiased estimates of the population mean and variance.

OCR S2 2011 January Q2

6 marks Standard +0.3

2 The random variable $H$ has the distribution $\mathrm { N } \left( \mu , 5 ^ { 2 } \right)$. The mean of a sample of $n$ observations of $H$ is denoted by $\bar { H }$. It is given that $\mathrm { P } ( \bar { H } > 53.28 ) = 0.0250$ and $\mathrm { P } ( \bar { H } < 51.65 ) = 0.0968$, both correct to 4 decimal places. Find the values of $\mu$ and $n$.

OCR S2 2011 January Q3

6 marks Moderate -0.8

3 The probability that a randomly chosen PPhone has a faulty casing is 0.0228 . A random sample of 200 PPhones is obtained. Use a suitable approximation to find the probability that the number of PPhones in the sample with a faulty casing is 2 or fewer. Justify your approximation.

OCR S2 2011 January Q4

7 marks Standard +0.3

4 The continuous random variable $X$ has mean $\mu$ and standard deviation 45. A significance test is to be carried out of the null hypothesis $\mathrm { H } _ { 0 } : \mu = 230$ against the alternative hypothesis $\mathrm { H } _ { 1 } : \mu \neq 230$, at the $1 \%$ significance level. A random sample of size 50 is obtained, and the sample mean is found to be 213.4.

Carry out the test.
Explain whether it is necessary to use the Central Limit Theorem in your test.

OCR S2 2011 January Q5

7 marks Standard +0.3

5 A temporary job is advertised annually. The number of applicants for the job is a random variable which is known from many years' experience to have a distribution $\operatorname { Po } ( 12 )$. In 2010 there were 19 applicants for the job. Test, at the 10\% significance level, whether there is evidence of an increase in the mean number of applicants for the job.

OCR S2 2011 January Q6

10 marks Standard +0.3

6 The number of randomly occurring events in a given time interval is denoted by $R$. In order that $R$ is well modelled by a Poisson distribution, it is necessary that events occur independently.

Let $R$ represent the number of customers dining at a restaurant on a randomly chosen weekday lunchtime. Explain what the condition 'events occur independently' means in this context, and give a reason why it would probably not hold in this context. Let $D$ represent the number of tables booked at the restaurant on a randomly chosen day. Assume that $D$ can be well modelled by distribution $\operatorname { Po } ( 7 )$.
Find $\mathrm { P } ( D < 5 )$.
Use a suitable approximation to find the probability that, in five randomly chosen days, the total number of tables booked is greater than 40 .

OCR S2 2011 January Q7

10 marks Moderate -0.8

7 Two continuous random variables $S$ and $T$ have probability density functions $\mathrm { f } _ { S }$ and $\mathrm { f } _ { T }$ given respectively by $$\begin{aligned} & f _ { S } ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases} \\ & f _ { T } ( x ) = \begin{cases} b & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases} \end{aligned}$$ where $a$ and $b$ are constants.

Sketch on the same axes the graphs of $y = \mathrm { f } _ { S } ( x )$ and $y = \mathrm { f } _ { T } ( x )$.
Find the value of $a$.
Find $\mathrm { E } ( S )$.
A student gave the following description of the distribution of $T$ : "The probability that $T$ occurs is constant". Give an improved description, in everyday terms.

OCR S2 2011 January Q8

11 marks Moderate -0.3

8 A company has 3600 employees, of whom $22.5 \%$ live more than 30 miles from their workplace. A random sample of 40 employees is obtained.

Use a suitable approximation, which should be justified, to find the probability that more than 5 of the employees in the sample live more than 30 miles from their workplace.
Describe how to use random numbers to select a sample of 40 from a population of 3600 employees.

OCR S2 2011 January Q9

11 marks Standard +0.3

9 A pharmaceutical company is developing a new drug to treat a certain disease. The company will continue to develop the drug if the proportion $p$ of those who have the disease and show a substantial improvement after treatment is greater than 0.7 . The company carries out a test, at the $5 \%$ significance level, on a random sample of 14 patients who suffer from the disease.

Find the critical region for the test.
Given that 12 of the 14 patients in the sample show a substantial improvement, carry out the test.
Find the probability that the test results in a Type II error if in fact $p = 0.8$. RECOGNISING ACHIEVEMENT