Questions — OCR MEI (4455 questions)

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OCR MEI S2 2006 June Q1
18 marks Standard +0.3
1 A low-cost airline charges for breakfasts on its early morning flights. On average, \(10 \%\) of passengers order breakfast.
  1. Find the probability that, out of 8 randomly selected passengers, exactly 1 orders breakfast.
  2. Use a suitable Poisson approximating distribution to find the probability that the number of breakfasts ordered by 30 randomly selected passengers is
    (A) exactly 6,
    (B) at least 8 .
  3. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
  4. The aircraft carries 120 passengers and the flight is always full. Find the mean \(\mu\) and variance \(\sigma ^ { 2 }\) of a Normal approximating distribution suitable for modelling the total number of passengers on the flight who order breakfast.
  5. Use your Normal approximating distribution to calculate the probability that more than 15 breakfasts are ordered on a particular flight.
  6. The airline wishes to be at least \(99 \%\) certain that the plane will have sufficient breakfasts for all passengers who order them. Find the minimum number of breakfasts which should be carried on each flight.
OCR MEI S2 2006 June Q2
18 marks Standard +0.3
2 The head circumference of 3-year-old boys is known to be Normally distributed with mean 49.7 cm and standard deviation 1.6 cm .
  1. Find the probability that the head circumference of a randomly selected 3 -year-old boy will be
    (A) over 51.5 cm ,
    (B) between 48.0 and 51.5 cm .
  2. Four 3-year-old boys are selected at random. Find the probability that exactly one of them has head circumference between 48.0 and 51.5 cm .
  3. The head circumference of 3-year-old girls is known to be Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(60 \%\) of 3-year-old girls have head circumference below 49.0 cm and \(30 \%\) have head circumference below 47.5 cm , find the values of \(\mu\) and \(\sigma\). A nutritionist claims that boys who have been fed on a special organic diet will have a larger mean head circumference than other boys. A random sample of ten 3 -year-old boys who have been fed on this organic diet is selected. It is found that their mean head circumference is 50.45 cm .
  4. Using the null and alternative hypotheses \(\mathrm { H } _ { 0 } : \mu = 49.7 \mathrm {~cm} , \mathrm { H } _ { 1 } : \mu > 49.7 \mathrm {~cm}\), carry out a test at the \(10 \%\) significance level to examine the nutritionist's claim. Explain the meaning of \(\mu\) in these hypotheses. You may assume that the standard deviation of the head circumference of organically fed 3 -year-old boys is 1.6 cm .
OCR MEI S2 2006 June Q3
18 marks Standard +0.3
3 A student is investigating the relationship between the length \(x \mathrm {~mm}\) and circumference \(y \mathrm {~mm}\) of plums from a large crop. The student measures the dimensions of a random sample of 10 plums from this crop. Summary statistics for these dimensions are as follows. $$\begin{aligned} & \sum x = 4715 \quad \sum y = 13175 \quad \sum x ^ { 2 } = 2237725 \\ & \sum y ^ { 2 } = 17455825 \quad \sum x y = 6235575 \quad n = 10 \end{aligned}$$
  1. Calculate the sample product moment correlation coefficient.
  2. Carry out a hypothesis test at the \(5 \%\) significance level to determine whether there is any correlation between length and circumference of plums from this crop. State your hypotheses clearly, defining any symbols which you use.
  3. (A) Explain the meaning of a 5\% significance level.
    (B) State one advantage and one disadvantage of using a \(1 \%\) significance level rather than a \(5 \%\) significance level in a hypothesis test. The student decides to take another random sample of 10 plums. Using the same hypotheses as in part (ii), the correlation coefficient for this second sample is significant at the \(5 \%\) level. The student decides to ignore the first result and concludes that there is correlation between the length and circumference of plums in the crop.
  4. Comment on the student's decision to ignore the first result. Suggest a better way in which the student could proceed.
OCR MEI S2 2006 June Q4
18 marks Standard +0.3
4 A survey of a random sample of 250 people is carried out. Their musical preferences are categorized as pop, classical or jazz. Their ages are categorized as under 25, 25 to 50, or over 50. The results are as follows.
\multirow{2}{*}{}Musical preference\multirow{2}{*}{Row totals}
PopClassicalJazz
\multirow{3}{*}{Age group}Under 2557151284
25-5043212185
Over 5022322781
Column totals1226860250
  1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between musical preference and age group. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
  2. Discuss briefly how musical preferences vary between the age groups, as shown by the contributions to the test statistic.
OCR MEI S2 2007 June Q1
19 marks Standard +0.3
1 The random variable \(X\) represents the time taken in minutes for a haircut at a barber's shop. \(X\) is Normally distributed with mean 11 and standard deviation 3 .
  1. Find \(\mathrm { P } ( X < 10 )\).
  2. Find the probability that exactly 3 out of 8 randomly selected haircuts take less than 10 minutes.
  3. Use a suitable approximating distribution to find the probability that at least 50 out of 100 randomly selected haircuts take less than 10 minutes. A new hairdresser joins the shop. The shop manager suspects that she takes longer on average than the other staff to do a haircut. In order to test this, the manager records the time taken for 25 randomly selected cuts by the new hairdresser. The mean time for these cuts is 12.34 minutes. You should assume that the time taken by the new hairdresser is Normally distributed with standard deviation 3 minutes.
  4. Write down suitable null and alternative hypotheses for the test.
  5. Carry out the test at the \(5 \%\) level.
OCR MEI S2 2007 June Q2
19 marks Standard +0.3
2 A medical student is trying to estimate the birth weight of babies using pre-natal scan images. The actual weights, \(x \mathrm {~kg}\), and the estimated weights, \(y \mathrm {~kg}\), of ten randomly selected babies are given in the table below.
\(x\)2.612.732.872.963.053.143.173.243.764.10
\(y\)3.22.63.53.12.82.73.43.34.44.1
  1. Calculate the value of Spearman's rank correlation coefficient.
  2. Carry out a hypothesis test at the \(5 \%\) level to determine whether there is positive association between the student's estimates and the actual birth weights of babies in the underlying population.
  3. Calculate the value of the product moment correlation coefficient of the sample. You may use the following summary statistics in your calculations: $$\Sigma x = 31.63 , \quad \Sigma y = 33.1 , \quad \Sigma x ^ { 2 } = 101.92 , \quad \Sigma y ^ { 2 } = 112.61 , \quad \Sigma x y = 106.51 .$$
  4. Explain why, if the underlying population has a bivariate Normal distribution, it would be preferable to carry out a hypothesis test based on the product moment correlation coefficient. Comment briefly on the significance of the product moment correlation coefficient in relation to that of Spearman's rank correlation coefficient.
OCR MEI S2 2007 June Q3
16 marks Standard +0.3
3 The number of calls received at an office per 5 minutes is modelled by a Poisson distribution with mean 3.2.
  1. Find the probability of
    (A) exactly one call in a 5 -minute period,
    (B) at least 6 calls in a 5 -minute period.
  2. Find the probability of
    (A) exactly one call in a 1 -minute period,
    (B) exactly one call in each of five successive 1-minute periods.
  3. Use a suitable approximating distribution to find the probability of at most 45 calls in a period of 1 hour. Two assumptions required for a Poisson distribution to be a suitable model are that calls arrive
OCR MEI S2 2007 June Q4
18 marks Standard +0.3
4 The sexes and ages of a random sample of 300 runners taking part in marathons are classified as follows.
ObservedSex\multirow{2}{*}{Row totals}
\cline { 3 - 4 }MaleFemale
\multirow{3}{*}{
Age
group
}		Under 407054124
\cline { 2 - 4 }\(40 - 49\)7636112
\cline { 2 - 5 }50 and over521264
Column totals198102300
  1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between age group and sex. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
  2. Does your analysis support the suggestion that women are less likely than men to enter marathons as they get older? Justify your answer. For marathons in general, on average \(3 \%\) of runners are 'Female, 50 and over'. The random variable \(X\) represents the number of 'Female, 50 and over' runners in a random sample of size 300.
  3. Use a suitable approximating distribution to find \(\mathrm { P } ( X \geqslant 12 )\).
OCR MEI S2 2008 June Q1
18 marks Standard +0.3
1 A researcher believes that there is a negative correlation between money spent by the government on education and population growth in various countries. A random sample of 48 countries is selected to investigate this belief. The level of government spending on education \(x\), measured in suitable units, and the annual percentage population growth rate \(y\), are recorded for these countries. Summary statistics for these data are as follows. $$\Sigma x = 781.3 \quad \Sigma y = 57.8 \quad \Sigma x ^ { 2 } = 14055 \quad \Sigma y ^ { 2 } = 106.3 \quad \Sigma x y = 880.1 \quad n = 48$$
  1. Calculate the sample product moment correlation coefficient.
  2. Carry out a hypothesis test at the \(5 \%\) significance level to investigate the researcher's belief. State your hypotheses clearly, defining any symbols which you use.
  3. State the distributional assumption which is necessary for this test to be valid. Explain briefly how a scatter diagram may be used to check whether this assumption is likely to be valid.
  4. A student suggests that if the variables are negatively correlated then population growth rates can be reduced by increasing spending on education. Explain why the student may be wrong. Discuss an alternative explanation for the correlation.
  5. State briefly one advantage and one disadvantage of using a smaller sample size in this investigation.
OCR MEI S2 2008 June Q2
18 marks Standard +0.3
2 A public water supply contains bacteria. Each day an analyst checks the water quality by counting the number of bacteria in a random sample of 5 ml of water. Throughout this question, you should assume that the bacteria occur randomly at a mean rate of 0.37 bacteria per 5 ml of water.
  1. Use a Poisson distribution to
    (A) find the probability that a 5 ml sample contains exactly 2 bacteria,
    (B) show that the probability that a 5 ml sample contains more than 2 bacteria is 0.0064 .
  2. The month of September has 30 days. Find the probability that during September there is at most one day when a 5 ml sample contains more than 2 bacteria. The daily 5 ml sample is the first stage of the quality control process. The remainder of the process is as follows.
OCR MEI S2 2008 June Q3
18 marks Moderate -0.3
3 A company has a fleet of identical vans. Company policy is to replace all of the tyres on a van as soon as any one of them is worn out. The random variable \(X\) represents the number of miles driven before the tyres on a van are replaced. \(X\) is Normally distributed with mean 27500 and standard deviation 4000.
  1. Find \(\mathrm { P } ( X > 25000 )\).
  2. 10 vans in the fleet are selected at random. Find the probability that the tyres on exactly 7 of them last for more than 25000 miles.
  3. The tyres of \(99 \%\) of vans last for more than \(k\) miles. Find the value of \(k\). A tyre supplier claims that a different type of tyre will have a greater mean lifetime. A random sample of 15 vans is fitted with these tyres. For each van, the number of miles driven before the tyres are replaced is recorded. A hypothesis test is carried out to investigate the claim. You may assume that these lifetimes are also Normally distributed with standard deviation 4000.
  4. Write down suitable null and alternative hypotheses for the test.
  5. For the 15 vans, it is found that the mean lifetime of the tyres is 28630 miles. Carry out the test at the \(5 \%\) level.
OCR MEI S2 2008 June Q4
18 marks Standard +0.3
4 A student is investigating whether there is any association between the species of shellfish that occur on a rocky shore and where they are located. A random sample of 160 shellfish is selected and the numbers of shellfish in each category are summarised in the table below.
Location
\cline { 3 - 5 } \multicolumn{2}{|c|}{}ExposedShelteredPool
\multirow{3}{*}{Species}Limpet243216
\cline { 2 - 5 }Mussel24113
\cline { 2 - 5 }Other52223
  1. Write down null and alternative hypotheses for a test to examine whether there is any association between species and location. The contributions to the test statistic for the usual \(\chi ^ { 2 }\) test are shown in the table below.
    ContributionLocation
    \cline { 3 - 5 }ExposedShelteredPool
    \multirow{3}{*}{Species}Limpet0.00090.25850.4450
    \cline { 2 - 5 }Mussel10.34721.27564.8773
    \cline { 2 - 5 }Other8.07190.14027.4298
    The sum of these contributions is 32.85 .
  2. Calculate the expected frequency for mussels in pools. Verify the corresponding contribution 4.8773 to the test statistic.
  3. Carry out the test at the \(5 \%\) level of significance, stating your conclusion clearly.
  4. For each species, comment briefly on how its distribution compares with what would be expected if there were no association.
  5. If 3 of the 160 shellfish are selected at random, one from each of the 3 types of location, find the probability that all 3 of them are limpets.
OCR MEI C1 Q1
3 marks Easy -1.8
1 Solve the inequality \(\frac { 4 x - 5 } { 7 } > 2 x + 1\).
OCR MEI C1 Q2
3 marks Moderate -0.8
2 Solve the inequality \(3 x ^ { 2 } + 10 x + 3 > 0\).
OCR MEI C1 Q3
4 marks Moderate -0.8
3 Solve the inequality \(5 x ^ { 2 } - 28 x - 12 \leqslant 0\).
OCR MEI C1 Q4
4 marks Easy -1.8
4 Solve the following inequality. $$\frac { 2 x + 1 } { 5 } < \frac { 3 x + 4 } { 6 }$$
OCR MEI C1 Q5
3 marks Easy -1.8
5 Solve the inequality \(6 ( x + 3 ) > 2 x + 5\).
OCR MEI C1 Q6
2 marks Easy -1.8
6 Solve the inequality \(5 - 2 x < 0\).
OCR MEI C1 Q7
5 marks Easy -1.2
7 Solve the following inequalities.
  1. \(2 ( 1 - x ) > 6 x + 5\)
  2. \(( 2 x - 1 ) ( x + 4 ) < 0\)
OCR MEI C1 Q8
3 marks Easy -1.8
8 Solve the inequality \(\frac { 5 x - 3 } { 2 } < x + 5\).
OCR MEI C1 Q9
2 marks Easy -1.2
9 Solve the inequality \(x ( x - 6 ) > 0\).
OCR MEI C1 Q10
3 marks Easy -1.8
10 Solve the inequality \(7 - x < 5 x - 2\).
OCR MEI C1 Q11
2 marks Easy -1.8
11 Solve the inequality \(3 x - 1 > 5 - x\).
OCR MEI C1 Q12
3 marks Easy -1.8
12 Solve the inequality \(1 - 2 x < 4 + 3 x\).
OCR MEI C1 Q13
4 marks Easy -1.2
13 Solve the inequality \(x ^ { 2 } + 2 x < 3\).