Edexcel Paper 3 (Paper 3) Specimen

1. Kaff coffee is sold in packets. A seller measures the masses of the contents of a random sample of 90 packets of Kaff coffee from her stock. The results are shown in the table below.

$$\text { (You may use } \sum \mathrm { fy } ^ { 2 } = 5644 \text { 171.75) }$$ A histogram is drawn and the class \(245 \leq w < 248\) is represented by a rectangle of width 1.2 cm and height 10 cm .

1. Calculate the width and the height of the rectangle representing the class \(255 \leq w < 260\).
2. Use linear interpolation to estimate the median mass of the contents of a packet of Kaff coffee to 1 decimal place.
3. Estimate the mean and the standard deviation of the mass of the contents of a packet of Kaff coffee to 1 decimal place. The seller claims that the mean mass of the contents of the packets is more than the stated mass. Given that the stated mass of the contents of a packet of Kaff coffee is 250 g and the actual standard deviation of the contents of a packet of Kaff coffee is 4 g ,
4. test, using a 5\% level of significance, whether or not the seller's claim is justified. State your hypotheses clearly.
   (You may assume that the mass of the contents of a packet is normally distributed.)
5. Using your answers to parts (b) and (c), comment on the assumption that the mass of the contents of a packet is normally distributed.
   (Total 14 marks)

2. A researcher believes that there is a linear relationship between daily mean temperature and daily total rainfall. The 7 places in the northern hemisphere from the large data set are used. The mean of the daily mean temperatures, \(t ^ { \circ } \mathrm { C }\), and the mean of the daily total rainfall, \(s \mathrm {~mm}\), for the month of July in 2015 are shown on the scatter diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{565bfa73-8095-4242-80b6-cd47aaff6a31-03_844_1339_497_372}

1. With reference to the scatter diagram, explain why a linear regression model may not be suitable for the relationship between \(t\) and s .
   (1) The researcher calculated the product moment correlation coefficient for the 7 places and obtained \(r = 0.658\).
2. Stating your hypotheses clearly, test at the \(10 \%\) level of significance, whether or not the product moment correlation coefficient for the population is greater than zero.
   (3)
3. Using your knowledge of the large data set, suggest the names of the 2 places labelled \(G\) and \(H\).
   (1)
4. Using your knowledge from the large data set, and with reference to the locations of the two places labelled \(G\) and \(H\), give a reason why these places have the highest temperatures in July.
   (2)
5. Suggest how you could make better use of the large data set to investigate the relationship between daily mean temperature and daily total rainfall.
   (1)
   (Total 7 marks)

3. For a particular type of bulb, \(36 \%\) grow into plants with blue flowers and the remainder grow into plants with white flowers. Bulbs are sold in mixed bags of 40 Russell selects a random sample of 5 bags of bulbs.

1. Find the probability that fewer than 2 of these bags will contain more bulbs that grow into plants with blue flowers than grow into plants with white flowers.
   (4) Maggie takes a random sample of \(n\) bulbs.
   Using a normal approximation, the probability that more than 244 of these \(n\) bulbs will grow into blue flowers is 0.0521 to 4 decimal places.
2. Find the value of \(n\).
   (6)
   (Total 10 marks)

4. The Venn diagram shows the probabilities of students' lunch boxes containing a drink, sandwiches and a chocolate bar.
\includegraphics[max width=\textwidth, alt={}, center]{565bfa73-8095-4242-80b6-cd47aaff6a31-05_655_899_392_484}
\(D\) is the event that a lunch box contains a drink, \(S\) is the event that a lunch box contains sandwiches,
\(C\) is the event that a lunch box contains a chocolate bar, \(u , v\) and \(w\) are probabilities.

1. Write down \(\mathrm { P } \left( S \cap D ^ { \prime } \right)\). One day, 80 students each bring in a lunch box.
   Given that all 80 lunch boxes contain sandwiches and a drink,
2. estimate how many of these 80 lunch boxes will contain a chocolate bar. Given that the events \(S\) and \(C\) are independent and that \(\mathrm { P } ( D \mid C ) = \frac { 14 } { 15 }\),
3. calculate the value of \(u\), the value of \(v\) and the value of \(w\).
   (7)
   (Total 11 marks)

5. The lifetimes of batteries sold by company \(X\) are normally distributed, with mean 150 hours and standard deviation 25 hours. A box contains 12 batteries from company \(X\).

1. Find the expected number of these batteries that have a lifetime of more than 160 hours. The lifetimes of batteries sold by company \(Y\) are normally distributed, with mean 160 hours and \(80 \%\) of these batteries have a lifetime of less than 180 hours.
2. Find the standard deviation of the lifetimes of batteries from company \(Y\). Both companies sell their batteries for the same price.
3. State which company you would recommend. Give reasons for your answer.