SPS SPS SM Statistics (SPS SM Statistics) 2023 January

1.

1. Joseph drew a histogram to show information about one Local Authority. He used data from the "Age structure by LA 2011" tab in the large data set. The table shows an extract from the data that he used.

   Age group	0 to 4
   Frequency	2143

   Joseph used a scale of \(1 \mathrm {~cm} = 1000\) units on the frequency density axis. Calculate the height of the histogram block for the 0 to 4 class.
2. Magdalene wishes to draw a statistical diagram to illustrate some of the data from the "Method of travel by LA 2011" tab in the large data set. State why she cannot draw a histogram.

2. Jane conducted a survey. She chose a sample of people from three towns, A, B and C. She noted the following information. 400 people were chosen.
230 people were adults.
55 adults were from town A .
65 children were from town A .
35 children were from town B .
150 people were from town \(B\).

1. In the Printed Answer Booklet, complete the two-way frequency table.

   \cline { 2 - 4 } \multicolumn{1}{c|}{}	Town
   \cline { 2 - 4 } \multicolumn{1}{c|}{}	A	B	C	Total
   adult
   child
   Total

2. One of the people is chosen at random.
   1. Find the probability that this person is an adult from town A .
   2. Given that the person is from town A , find the probability that the person is an adult. For another survey, Jane wanted to choose a random sample from the 820 students living in a particular hostel. She numbered the students from 1 to 820 and then generated some random numbers on her calculator. The random numbers were 0.114287562 and 0.081859817 .
      Jane's friend Kareem used these figures to write down the following sample of five student numbers. 114, 142, 428, 287 and 756
      Jane used the same figures to write down the following sample of five student numbers.
      \(114,287,562,81\) and 817
3. 1. State, with a reason, which one of these samples is not random.
   2. Explain why Jane omitted the number 859 from her sample.

3. Pierre is a chef. He claims that \(90 \%\) of his customers are satisfied with his cooking. Yvette suspects that Pierre is over-confident about the level of satisfaction amongst his customers. She talks to a random sample of 15 of Pierre's customers, and finds that 11 customers say that they are satisfied. She then performs a hypothesis test. Carry out the test at the 5\% significance level.

4.

1. The masses, in grams, of plums of a certain kind have the distribution \(\mathrm { N } ( 55,18 )\). The heaviest \(5 \%\) of plums are classified as extra large. Find the minimum mass of extra large plums.
2. The masses, in grams, of apples of a certain kind have the distribution \(\mathrm { N } \left( 67 , \sigma ^ { 2 } \right)\). It is given that half of the apples have masses between 62 g and 72 g . Determine \(\sigma\).

5. Each member of a group of 27 people was timed when completing a puzzle.
The time taken, \(x\) minutes, for each member of the group was recorded.
These times are summarised in the following box and whisker plot.
\includegraphics[max width=\textwidth, alt={}, center]{f03113c4-039e-4ead-9588-b4b83fb7eea9-08_381_1557_504_264}

1. Find the range of the times.
2. Find the interquartile range of the times. For these 27 people \(\sum x = 607.5\) and \(\sum x ^ { 2 } = 17623.25\)
3. calculate the mean time taken to complete the puzzle,
4. calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.
5. State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in \(a\) minutes and \(b\) minutes respectively, where \(a > b\).
   When their times are included with the data of the other 27 people
   • the median time increases
   • the mean time does not change
   • Suggest a possible value for \(a\) and a possible value for \(b\), explaining how your values satisfy the above conditions.
   • Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).

6. The level, in grams per millilitre, of a pollutant at different locations in a certain river is denoted by the random variable \(X\), where \(X\) has the distribution \(\mathrm { N } ( \mu , 0.0000409 )\). In the past the value of \(\mu\) has been 0.0340 .
This year the mean level of the pollutant at 50 randomly chosen locations was found to be 0.0325 grams per millilitre. Test, at the \(5 \%\) significance level, whether the mean level of pollutant has changed.

7. A large college produces three magazines.
One magazine is about green issues, one is about equality and one is about sports. A student at the college is selected at random and the events \(G , E\) and \(S\) are defined as follows
\(G\) is the event that the student reads the magazine about green issues \(E\) is the event that the student reads the magazine about equality \(S\) is the event that the student reads the magazine about sports The Venn diagram, where \(p , q , r\) and \(t\) are probabilities, gives the probability for each subset.
\includegraphics[max width=\textwidth, alt={}, center]{f03113c4-039e-4ead-9588-b4b83fb7eea9-12_533_903_790_548}

1. Find the proportion of students in the college who read exactly one of these magazines. No students read all three magazines and \(\mathrm { P } ( G ) = 0.25\)
2. Find
   1. the value of \(p\)
   2. the value of \(q\) Given that \(\mathrm { P } ( S \mid E ) = \frac { 5 } { 12 }\)
3. find
   1. the value of \(r\)
   2. the value of \(t\)
4. Determine whether or not the events ( \(S \cap E ^ { \prime }\) ) and \(G\) are independent. Show your working clearly. END OF TEST