2. Chi wanted to summarise the scores of the 39 competitors in a village quiz. He started to produce the following stem and leaf diagram \( \qquad \) Key: \( 2 \mid 5 \) is a score of 25 He did not complete the stem and leaf diagram but instead produced the following box plot. Chi defined an outlier as a value that is Score \[ \text { greater than } Q_{3}+1.5 \times\left(Q_{3}-Q_{1}\right) \] or less than \( Q_{1}-1.5 \times\left(Q_{3}-Q_{1}\right) \) (a) Find (i) the interquartile range (ii) the range. (2) (b) Describe, giving a reason, the skewness of the distribution of scores. (2) Albert and Beth asked for their scores to be checked. Albert's score was changed from 25 to 37 Beth's score was changed from 54 to 60 (c) On the grid on page 5, draw an updated box plot. Show clearly any calculations that you used. Some of the competitors complained that the questions were biased towards the younger generation. The product moment correlation coefficient between the age of the competitors and their score in the quiz is -0.187 (d) State, giving a reason, whether or not the complaint is supported by this statistic.
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We have a stem and leaf diagram and a box plot for the scores of 39 competitors. We need to find the interquartile range (IQR), the range, describe the skewness, update the box plot after score changes, and interpret the correlation coefficient. Show more…
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2. (a) The stem and leaf diagram shows the number of deliveries made by Francis each day for 24 days Stem Leaf Total Key: 10|9 means 109 10 8 9 (2) 11 0 3 6 6 6 8 8 9 9 9 9 (11) 12 4 5 5 5 5 5 5 8 (8) 13 a b c (3) where a, b and c are positive integers with a < b < c An outlier is defined as any value greater than 1.5 x interquartile range above the upper quartile. Given that there is only one outlier for these data, (i) show that c = 9. The number of deliveries made by Francis each day is represented by d. The data in the stem and leaf diagram are coded using x = d - 125 and the following summary statistics are obtained sum x = -96 and sum (x - x_bar)^2 = 1306 (ii) Find the mean number of deliveries. (iii) Find the standard deviation of the number of deliveries. One of these 24 days is selected at random. The random variable D represents the number of deliveries made by Francis on this day. The random variable X = D - 125. (iv) Find P(D > 118 | X < 0).
Kirsty G.
1. A researcher is investigating the growth of two types of tree, Birch and Maple. The height, to the nearest cm, a seedling grows in one year is recorded for 35 Birch trees and 32 Maple trees. The results are summarised in the back-to-back stem and leaf diagram below. Birch Maple Totals (2) 9 8 2 5 7 7 8 9 (5) (8) 9 9 9 6 5 3 1 1 3 0 2 6 6 8 9 9 (7) (9) 9 8 8 7 6 3 1 1 1 4 1 1 1 k 7 8 (6) (9) 7 7 7 5 4 3 2 1 0 5 0 1 2 3 4 4 4 (7) (3) 7 6 5 6 3 4 6 (3) (3) 6 5 4 7 0 7 (2) (1) 5 8 0 0 (2) Key: 5 | 6 | 3 means 65 cm for a Birch tree and 63 cm for a Maple tree The median height that these Maple trees grow in one year is 45 cm. (a) Find the value of k, used in the stem and leaf diagram. (1) (b) Find the lower quartile and the upper quartile of the height grown in one year for these Birch trees. (2) The researcher defines an outlier as an observation that is greater than Q3 + 1.5 x (Q3 - Q1) or less than Q1 - 1.5 x (Q3 - Q1) (c) Show that there is only one outlier amongst the Birch trees. (2) The grid on page 3 shows a box plot for the heights that the Maple trees grow in one year. (d) On the same grid draw a box plot for the heights that the Birch trees grow in one year. (4) (e) Comment on any difference in the distributions of the growth of these Birch trees and the growth of these Maple trees. State the values of any statistics you have used to support your comment. (1) The researcher realises he has missed out 4 pieces of data for the Maple trees. The heights each seedling grows in one year, to the nearest cm, in ascending order, for these 4 Maple trees are 27 cm, a cm, 48 cm, 2a cm. Given that there is no change to the box plot for the Maple trees given on page 3 (f) find the range of possible values for a Show your working clearly. (3)
Kenny M.
A survey was conducted of 130 purchasers of new BMW 3 series cars, 130 purchasers of new BMW 5 series cars, and 130 purchasers of new BMW 7 series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results. (FIGURE CAN'T COPY) a. In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected for that car series. b. Which group is most likely to have an outlier? Explain how you determined that. c. Compare the three box plots. What do they imply about the age of purchasing a BMW from the series when compared to each other? d. Look at the BMW 5 series. Which quarter has the smallest spread of data? What is the spread? e. Look at the BMW 5 series. Which quarter has the largest spread of data? What is the spread? f. Look at the BMW 5 series. Estimate the interquartile range (IQR). g. Look at the BMW 5 series. Are there more data in the interval 31 to 38 or in the interval 45 to 55? How do you know this? h. Look at the BMW 5 series. Which interval has the fewest data in it? How do you know this? $$\begin{array}{l}{\text { i. } 31-35} \\ {\text { ii. } 38-41} \\ {\text { iii. } 41-64}\end{array}$$
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