SPS SPS SM Statistics (SPS SM Statistics) 2025 January

1. Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.

• \(A\) is the event that Isobel's parents watch a match.
• \(B\) is the event that Isobel scores in a match.

You are given that \(\mathrm { P } ( B \mid A ) = \frac { 3 } { 7 }\) and \(\mathrm { P } ( A ) = \frac { 7 } { 10 }\).

1. Calculate \(\mathrm { P } ( A \cap B )\). The probability that Isobel does not score and her parents do not attend is 0.1 .
2. Draw a Venn diagram showing the events \(A\) and \(B\), and mark in the probability corresponding to each of the regions of your diagram.
3. Are events \(A\) and \(B\) independent? Give a reason for your answer.
4. By comparing \(\mathrm { P } ( B \backslash A )\) with \(\mathrm { P } ( B )\), explain why Isobel should ask her parents not to attend.
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2. On average, \(25 \%\) of the packets of a certain kind of soup contain a voucher. Kim buys one packet of soup each week for 12 weeks. The number of vouchers she obtains is denoted by \(X\).

1. State two conditions needed for \(X\) to be modelled by the distribution \(\mathrm { B } ( 12,0.25 )\). In the rest of this question you should assume that these conditions are satisfied.
2. Find \(\mathrm { P } ( X \leqslant 6 )\). In order to claim a free gift, 7 vouchers are needed.
3. Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.
4. Find the probability that Kim will be able to claim a free gift in the 12th week but not before.
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3. The continuous random variable \(T\) has mean \(\mu\) and standard deviation \(\sigma\). It is known that \(\mathrm { P } ( T < 140 ) = 0.01\) and \(\mathrm { P } ( T < 300 ) = 0.8\).

1. Assuming that \(T\) is normally distributed, calculate the values of \(\mu\) and \(\sigma\). In fact, \(T\) represents the time, in minutes, taken by a randomly chosen runner in a public marathon, in which about \(10 \%\) of runners took longer than 400 minutes.
2. State with a reason whether the mean of \(T\) would be higher than, equal to, or lower than the value calculated in part (i).
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4. The table shows information about the time, \(t\) minutes correct to the nearest minute, taken by 50 people to complete a race.

1. In a histogram illustrating the data, the height of the block for the \(31 \leqslant t \leqslant 35\) class is 5.6 cm . Find the height of the block for the \(28 \leqslant t \leqslant 30\) class. (There is no need to draw the histogram.)
2. The data in the table are used to estimate the median time. State, with a reason, whether the estimated median time is more than 33 minutes, less than 33 minutes or equal to 33 minutes.
3. Calculate estimates of the mean and standard deviation of the data.
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5. The masses, \(m\) grams, of 52 apples of a certain variety were found and summarised as follows. $$n = 52 \quad \Sigma ( m - 150 ) = - 182 \quad \Sigma ( m - 150 ) ^ { 2 } = 1768$$ Calculate the variance and thus the exact value of \(\sum m ^ { 2 }\)
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6. A television company believes that the proportion of households that can receive Channel C is 0.35 .

1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the \(2.5 \%\) significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35 .
2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working.
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7. The table shows information, derived from the 2011 UK census, about the percentage of employees who used various methods of travel to work in four Local Authorities. One of the Local Authorities is a London borough and two are metropolitan boroughs, not in London.

1. Which one of the Local Authorities is a London borough? Give a reason for your answer.
2. Which two of the Local Authorities are metropolitan boroughs outside London? In each case give a reason for your answer.
3. Describe one difference between the public transport available in the two metropolitan boroughs, as suggested by the table.
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8. 80 randomly chosen people are asked to estimate a time interval of 60 seconds without using a watch or clock. The mean of the 80 estimates is 58.9 seconds. Previous evidence shows that the population standard deviation of such estimates is 5.0 seconds. Test, at the \(5 \%\) significance level, whether there is evidence that people tend to underestimate the time interval.
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