SPS SPS SM Statistics (SPS SM Statistics) 2024 September

1. The histogram shows information about the lengths, \(l\) centimetres, of a sample of worms of a certain species.
\includegraphics[max width=\textwidth, alt={}, center]{c5ea8584-939f-4627-8f81-bac60233d9a3-04_913_1303_392_175} The number of worms in the sample with lengths in the class \(3 \leqslant l < 4\) is 30 .

1. Find the number of worms in the sample with lengths in the class \(0 \leqslant l < 2\).
2. Find an estimate of the number of worms in the sample with lengths in the range \(4.5 \leqslant l < 5.5\).
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2. A factory buys \(10 \%\) of its components from supplier \(A , 30 \%\) from supplier \(B\) and the rest from supplier \(C\). It is known that \(6 \%\) of the components it buys are faulty. Of the components bought from supplier \(A , 9 \%\) are faulty and of the components bought from supplier \(B , 3 \%\) are faulty.

1. Find the percentage of components bought from supplier \(C\) that are faulty. A component is selected at random.
2. Explain why the event "the component was bought from supplier \(B\) " is not statistically independent from the event "the component is faulty".
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3. The discrete random variable \(X\) takes values \(1,2,3,4\) and 5 , and its probability distribution is defined as follows. $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 1
\frac { 1 } { 2 } \mathrm { P } ( X = x - 1 ) & x = 2,3,4,5
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac { 16 } { 31 }\). The discrete probability distribution for \(X\) is given in the table.

   \(x\)	1	2	3	4	5
   \(\mathrm { P } ( X = x )\)	\(\frac { 16 } { 31 }\)	\(\frac { 8 } { 31 }\)	\(\frac { 4 } { 31 }\)	\(\frac { 2 } { 31 }\)	\(\frac { 1 } { 31 }\)

2. Find the probability that \(X\) is odd. Two independent values of \(X\) are chosen, and their sum \(S\) is found.
3. Find the probability that \(S\) is odd.
4. Find the probability that \(S\) is greater than 8 , given that \(S\) is odd. Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable \(Y\) defined as follows. $$\mathrm { P } ( Y = y + 1 ) = \frac { 1 } { 2 } \mathrm { P } ( Y = y ) \quad \text { for all positive integers } y .$$
5. Find \(\mathrm { P } ( Y = 1 )\).
6. Give a reason why one of the variables, \(X\) or \(Y\), might be more appropriate as a model for the number of attempts that Sheila needs to start her car.
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4. The radar diagrams illustrate some population figures from the 2011 census results.
\includegraphics[max width=\textwidth, alt={}, center]{c5ea8584-939f-4627-8f81-bac60233d9a3-10_723_776_360_159}
\includegraphics[max width=\textwidth, alt={}, center]{c5ea8584-939f-4627-8f81-bac60233d9a3-10_725_775_358_1055} Each radius represents an age group, as follows: The distance of each dot from the centre represents the number of people in the relevant age group.

1. The scales on the two diagrams are different. State an advantage and a disadvantage of using different scales in order to make comparisons between the ages of people in these two Local Authorities.
2. Approximately how many people aged 45 to 59 were there in Liverpool?
3. State the main two differences between the age profiles of the two Local Authorities.
4. James makes the following claim.
   "Assuming that there are no significant movements of population either into or out of the two regions, the 2021 census results are likely to show an increase in the number of children in Liverpool and a decrease in the number of children in Rutland." Use the radar diagrams to give a justification for this claim.
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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 Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students.
\includegraphics[max width=\textwidth, alt={}, center]{c5ea8584-939f-4627-8f81-bac60233d9a3-16_542_883_459_148} A student is chosen at random from the 100 students. Then another student is chosen from the remaining students. Find the probability that the first student studies History and the second student studies Geography but not Psychology.
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