Questions — OCR S3 (139 questions)

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OCR S3 2013 June Q2
2 In order to estimate the total number of rabbits in a certain region, a random sample of 500 rabbits is captured, marked and released. After two days a random sample of 250 rabbits is captured and 24 are found to be marked. It may be assumed that there is no change in the population during the two days.
  1. Estimate the total number of rabbits in the region.
  2. Calculate an approximate \(95 \%\) confidence interval for the population proportion of marked rabbits.
  3. Using your answer to part (ii), estimate a 95\% confidence interval for the total number of rabbits in the region.
OCR S3 2013 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{c4adc528-ae3f-4ea7-9420-d3e1068a85fe-2_524_796_1105_623} The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a x & 0 < x \leqslant 1
b ( 2 - x ) ^ { 2 } & 1 < x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants. The graph is shown in the above diagram.
  1. Find the values of \(a\) and \(b\).
  2. Find the value of \(\mathrm { E } \left( \frac { 1 } { X } \right)\).
OCR S3 2013 June Q4
4 A new computer was bought by a local council to search council records and was tested by an employee. She searched a random sample of 500 records and the sample mean search time was found to be 2.18 milliseconds and an unbiased estimate of variance was \(1.58 ^ { 2 }\) milliseconds \({ } ^ { 2 }\).
  1. Calculate a \(98 \%\) confidence interval for the population mean search time \(\mu\) milliseconds.
  2. It is required to obtain a sample mean time that differs from \(\mu\) by less than 0.05 milliseconds with probability 0.95 . Estimate the sample size required.
  3. State why it is unnecessary for the validity of your calculations that search time has a normal distribution.
OCR S3 2013 June Q5
5 The continuous random variable \(Y\) has probability density function given by $$\mathrm { f } ( y ) = \begin{cases} \ln ( y ) & 1 \leqslant y \leqslant \mathrm { e }
0 & \text { otherwise } \end{cases}$$
  1. Verify that this is a valid probability density function.
  2. Show that the (cumulative) distribution function of \(Y\) is given by $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 1
    y \ln y - y + 1 & 1 \leqslant y \leqslant \mathrm { e }
    1 & \text { otherwise } \end{cases}$$
  3. Verify that the upper quartile of \(Y\) lies in the interval [2.45, 2.46].
  4. Find the (cumulative) distribution function of \(X\) where \(X = \ln Y\).
OCR S3 2013 June Q6
6 A random sample of 80 students who had all studied Biology, Chemistry and Art at a college was each asked which they enjoyed most. The results, classified according to gender, are given in the table.
Subject
\cline { 2 - 5 }BiologyChemistryArt
\cline { 2 - 5 } GenderMale13411
\cline { 2 - 5 }Female3787
\cline { 2 - 5 }
\cline { 2 - 5 }
It is required to carry out a test of independence between subject most enjoyed and gender at the \(2 \frac { 1 } { 2 } \%\) significance level.
  1. Calculate the expected values for the cells.
  2. Explain why it is necessary to combine cells, and choose a suitable combination.
  3. Carry out the test.
OCR S3 2013 June Q7
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
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
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
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
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
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
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
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
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}