Questions — OCR S3 (141 questions)

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OCR S3 2013 June Q7
15 marks Standard +0.3
7 Two machines \(A\) and \(B\) both pack cartons in a factory. The mean packing times are compared by timing the packing of 10 randomly chosen cartons from machine \(A\) and 8 randomly chosen cartons from machine \(B\). The times, \(t\) seconds, taken to pack these cartons are summarised below.
Sample size\(\sum t\)\(\sum t ^ { 2 }\)
Machine \(A\)10221.44920.9
Machine \(B\)8199.24980.3
The packing times have independent normal distributions.
  1. Stating a necessary assumption, carry out a test, at the \(1 \%\) significance level, of whether the population mean packing times differ for the two machines.
  2. Find the largest possible value of the constant \(c\) for which there is evidence at the \(1 \%\) significance level that \(\mu _ { B } - \mu _ { A } > c\), where \(\mu _ { B }\) and \(\mu _ { A }\) denote the respective population mean packing times in seconds.
OCR S3 2016 June Q1
4 marks Standard +0.3
1 On a motorway, lorries pass an observation point independently and at random times. The mean number of lorries travelling north is 6 per minute and the mean number travelling south is 8 per minute. Find the probability that at least 16 lorries pass the observation point in a given 1 -minute period.
OCR S3 2016 June Q2
7 marks Standard +0.3
2 A random sample of 200 American voters were asked about which political party they supported and their attitude to a proposed new form of taxation. The voters' responses are summarised in the table. Attitude
\cline { 2 - 5 }In favourNeutralAgainst
\cline { 2 - 5 }Democrat581616
\cline { 2 - 5 } PartyIndependent25411
\cline { 2 - 5 }Republican172033
\cline { 2 - 5 }
\cline { 2 - 5 }
Carry out a \(\chi ^ { 2 }\) test, at the \(1 \%\) level of significance, to investigate whether there is an association between party supported and attitude to the proposed form of taxation.
OCR S3 2016 June Q3
8 marks Moderate -0.3
3
  1. A company packages butter. Of 50 randomly selected packs, 8 were found to have damaged wrappers. Find an approximate \(95 \%\) confidence interval for the proportion of packs with damaged wrappers.
  2. The mass of a pack has a normal distribution with standard deviation 8.5 g . In a random sample of 10 packs the masses, in g , are as follows. $$\begin{array} { l l l l l l l l l l } 220 & 225 & 218 & 223 & 224 & 220 & 229 & 228 & 226 & 228 \end{array}$$ Find a 99\% confidence interval for the mean mass of a pack.
OCR S3 2016 June Q4
9 marks Standard +0.3
4 A group of students were tested in geography before and after a fieldwork course. The marks of 10 randomly selected students are shown in the table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Mark before fieldwork19848499591929495469
Mark after fieldwork23988388683328535888
  1. Use a suitable \(t\)-test, at the \(5 \%\) level of significance, to test whether the students' performance has improved.
  2. State the necessary assumption in applying the test.
OCR S3 2016 June Q5
11 marks Standard +0.8
5 The independent random variables \(X\) and \(Y\) have distributions \(\mathrm { N } \left( 30 , \sigma ^ { 2 } \right)\) and \(\mathrm { N } \left( 20 , \sigma ^ { 2 } \right)\) respectively. The random variable \(a X + b Y\), where \(a\) and \(b\) are constants, has the distribution \(\mathrm { N } \left( 410,130 \sigma ^ { 2 } \right)\).
  1. Given that \(a\) and \(b\) are integers, find the value of \(a\) and the value of \(b\).
  2. Given that \(\mathrm { P } ( X > Y ) = 0.966\), find \(\sigma ^ { 2 }\).
OCR S3 2016 June Q6
11 marks Challenging +1.2
6 The masses at birth, in kg, of random samples of babies were recorded for each of the years 1970 and 2010. The table shows the sample mean and an unbiased estimate of the population variance for each year.
YearNo. of babies
Sample
mean
Unbiased estimate of
population variance
19702853.3030.2043
20102603.3520.2323
  1. A researcher tests the null hypothesis that babies born in 2010 are 0.04 kg heavier, on average, than babies born in 1970, against the alternative hypothesis that they are more than 0.04 kg heavier on average. Show that, at the \(5 \%\) level of significance, the null hypothesis is not rejected.
  2. Another researcher chooses samples of equal size, \(n\), for the two years. Using the same hypothesis as in part (i), she finds that the null hypothesis is rejected at the \(5 \%\) level of significance. Assuming that the sample means and unbiased estimates of population variance for the two years are as given in the table, find the smallest possible value of \(n\).
OCR S3 2016 June Q7
12 marks Standard +0.8
7 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 1 \\ a x ^ { 2 } & 1 < x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that \(a = \frac { 12 } { 31 }\).
  2. Find \(\mathrm { E } ( X )\). It is thought that the time taken by a student to complete a task can be well modelled by \(X\). The times taken by 992 randomly chosen students are summarised in the table, together with some of the expected frequencies.
    Time\(0 \leqslant x < 0.5\)\(0.5 \leqslant x < 1\)\(1 \leqslant x < 1.5\)\(1.5 \leqslant x \leqslant 2\)
    Observed frequency892279613
    Expected frequency690
  3. Find the other expected frequencies and test, at the \(5 \%\) level of significance, whether the data can be well modelled by \(X\).
OCR S3 2016 June Q8
10 marks Challenging +1.2
8 The radius, \(R\), of a sphere is a random variable with a continuous uniform distribution between 0 and 10 .
  1. Find the cumulative distribution function and probability density function of \(A\), the surface area of the sphere.
  2. Find \(\mathrm { P } ( \mathrm { A } \leqslant 200 \pi )\). \section*{END OF QUESTION PAPER}
OCR S3 2012 January Q1
6 marks Moderate -0.8
In a test of association of two factors, \(A\) and \(B\), a \(2 \times 2\) contingency table yielded \(5.63\) for the value of \(\chi^2\) with Yates' correction.
  1. State the null hypothesis and alternative hypothesis for the test. [1]
  2. State how Yates' correction is applied, and whether it increases or decreases the value of \(\chi^2\). [2]
  3. Carry out the test at the \(2\frac{1}{2}\%\) significance level. [3]
OCR S3 2012 January Q2
7 marks Standard +0.3
An investigation in 2007 into the incidence of tuberculosis (TB) in badgers in a certain area found that 42 out of a random sample of 190 badgers tested positive for TB. In 2010, 48 out of a random sample of 150 badgers tested positive for TB.
  1. Assuming that the population proportions of badgers with TB are the same in 2007 and 2010, obtain the best estimate of this proportion. [1]
  2. Carry out a test at the \(2\frac{1}{2}\%\) significance level of whether the population proportion of badgers with TB increased from 2007 to 2010. [6]
OCR S3 2012 January Q3
8 marks Standard +0.3
The continuous random variable \(U\) has a normal distribution with unknown mean \(\mu\) and known variance 1. A random sample of four observations of \(U\) gave the values \(3.9, 2.1, 4.6\) and \(1.4\).
  1. Calculate a \(90\%\) confidence interval for \(\mu\). [3]
  2. The probability that the sum of four random observations of \(U\) is less than 11 is denoted by \(p\). For each of the end points of the confidence interval in part (i) calculate the corresponding value of \(p\). [5]
OCR S3 2012 January Q4
10 marks Standard +0.3
\(X\) is a continuous random variable with the distribution N\((48.5, 12.5^2)\). The values of \(X\) are transformed to standardised values of \(Y\), using the equation \(Y = aX + b\), where \(a\) and \(b\) are constants with \(a > 0\).
  1. Find values of \(a\) and \(b\) for which the mean and standard deviation of \(Y\) are 40 and 10 respectively. [4]
  2. State the distribution of \(Y\). [1]
Two randomly chosen standardised values are denoted by \(Y_1\) and \(Y_2\).
  1. Calculate the probability that \(Y_2\) is at least 10 greater than \(Y_1\). [5]
OCR S3 2012 January Q5
10 marks Standard +0.3
A statistician suggested that the weekly sales \(X\) thousand litres at a petrol station could be modelled by the following probability density function. $$\text{f}(x) = \begin{cases} \frac{1}{40}(2x + 3) & 0 \leqslant x < 5, \\ 0 & \text{otherwise.} \end{cases}$$
  1. Show that, using this model, P\((a < X < a + 1) = \frac{a + 2}{20}\) for \(0 \leqslant a < 4\). [3]
Sales in 100 randomly chosen weeks gave the following grouped frequency table.
\(x\)\(0 \leqslant x < 1\)\(1 \leqslant x < 2\)\(2 \leqslant x < 3\)\(3 \leqslant x < 4\)\(4 \leqslant x < 5\)
Frequency1612183024
  1. Carry out a goodness of fit test at the \(10\%\) significance level of whether f\((x)\) fits the data. [7]
OCR S3 2012 January Q6
13 marks Standard +0.3
The continuous random variable \(Y\) has probability density function given by $$\text{f}(y) = \begin{cases} -\frac{1}{4}y & -2 < y < 0, \\ \frac{1}{4}y & 0 \leqslant y \leqslant 2, \\ 0 & \text{otherwise.} \end{cases}$$ Find
  1. the interquartile range of \(Y\), [4]
  2. Var\((Y)\), [5]
  3. E\((|Y|)\). [4]
OCR S3 2012 January Q7
18 marks Standard +0.3
The manufacturer's specification for batteries used in a certain electronic game is that the mean lifetime should be 32 hours. The manufacturer tests a random sample of 10 batteries made in Factory A, and the lifetimes (\(x\) hours) are summarised by \(n = 10\), \(\sum x = 289.0\) and \(\sum x^2 = 8586.19\). It may be assumed that the population of lifetimes has a normal distribution.
  1. Carry out a one-tail test at the \(5\%\) significance level of whether the specification is being met. [7]
  2. Justify the use of a one-tail test in this context. [1]
Batteries made with the same specification are also made in Factory B. The lifetimes of these batteries are also normally distributed. A random sample of 12 batteries from this factory was tested. The lifetimes are summarised by \(n = 12\), \(\sum x = 363.0\) and \(\sum x^2 = 11290.95\).
    1. State what further assumption must be made in order to test whether there is any difference in the mean lifetimes of batteries made at the two factories. Use the data to comment on whether this assumption is reasonable. [3]
    2. Carry out the test at the \(10\%\) significance level. [7]