Revision, normal distribution

Transcription

1 Revision, normal distribution 1a. [3 marks] The Brahma chicken produces eggs with weights in grams that are normally distributed about a mean of with a standard deviation of. The eggs are classified as small, medium, large or extra large according to their weight, as shown in the table below. Sketch a diagram of the distribution of the weight of Brahma chicken eggs. On your diagram, show clearly the boundaries for the classification of the eggs. 1b. [4 marks] An egg is chosen at random. Find the probability that the egg is(i) medium;(ii) extra large. 1c. [2 marks] There is a probability of that a randomly chosen egg weighs more than grams. Find. 1d. [2 marks] The probability that a Brahma chicken produces a large size egg is produce eggs each month.. Frank s Brahma chickens Calculate an estimate of the number of large size eggs produced by Frank s chickens each month. 1

2 1e. [3 marks] The selling price, in US dollars (USD), of each size is shown in the table below. probability that a Brahma chicken produces a small size egg is. The Estimate the monthly income, in USD, earned by selling the decimal places. eggs. Give your answer correct to two 2a. [1 mark] The heights of apple trees in an orchard are normally distributed with a mean of standard deviation of. and a Write down the probability that a randomly chosen tree has a height greater than. 2b. [1 mark] Write down the probability that a randomly chosen tree will be within 2 standard deviations of the mean of. 2c. [2 marks] Use your graphic display calculator to calculate the probability that a randomly chosen tree will have a height greater than. 2d. [2 marks] The probability that a particular tree is less than metres high is. Find the value of. 2

3 3a. [2 marks] A group of candidates sat a Chemistry examination and a Physics examination. The candidates marks in the Chemistry examination are normally distributed with a mean of and a standard deviation of. Draw a diagram that shows this information. 3b. [1 mark] Write down the probability that a randomly chosen candidate who sat the Chemistry examination scored at most 60 marks. 3c. [2 marks] Hee Jin scored 80 marks in the Chemistry examination. Find the probability that a randomly chosen candidate who sat the Chemistry examination scored more than Hee Jin. 3d. [2 marks] The candidates marks in the Physics examination are normally distributed with a mean of standard deviation of. Hee Jin also scored marks in the Physics examination. and a Find the probability that a randomly chosen candidate who sat the Physics examination scored less than Hee Jin. 3e. [2 marks] The candidates marks in the Physics examination are normally distributed with a mean of standard deviation of. Hee Jin also scored marks in the Physics examination. and a Determine whether Hee Jin s Physics mark, compared to the other candidates, is better than her mark in Chemistry. Give a reason for your answer. 3f. [3 marks] To obtain a grade A a candidate must be in the top examination. of the candidates who sat the Physics Find the minimum possible mark to obtain a grade A. Give your answer correct to the nearest integer. 3

4 4a. [2 marks] The daily January temperature of Cairns is normally distributed with a mean of 34 C and a standard deviation of 3. Calculate the probability that the temperature on a randomly chosen day in January is less than 39 C. 4b. [2 marks] Calculate the expected number of days in January that the temperature will be more than 39 C. 4c. [2 marks] On a randomly chosen day in January, the probability that the temperature is above C is 0.7. Find the value of. 5a. [2 marks] The weight,, of bags of rice follows a normal distribution with mean 1000 g and standard deviation 4 g. Find the probability that a bag of rice chosen at random weighs between 990 g and 1004 g. 5b. [2 marks] 95% of the bags of rice weigh less than grams. Find the value of. 5c. [2 marks] For a bag of rice chosen at random,. Find the value of. 4

5 6a. [2 marks] A factory makes metal bars. Their lengths are assumed to be normally distributed with a mean of 180 cm and a standard deviation of 5 cm. On the following diagram, shade the region representing the probability that a metal bar, chosen at random, will have a length less than 175 cm. 6b. [4 marks] A metal bar is chosen at random. (i) The probability that the length of the metal bar is less than 175 cm is equal to the probability that the length is greater than cm. Write down the value of. (ii) Find the probability that the length of the metal bar is greater than one standard deviation above the mean. 5

6 7a. [3 marks] Daniel grows apples and chooses at random a sample of 100 apples from his harvest. He measures the diameters of the apples to the nearest cm. The following table shows the distribution of the diameters. Using your graphic display calculator, write down the value of (i) the mean of the diameters in this sample; (ii) the standard deviation of the diameters in this sample. 7b. [3 marks] Daniel assumes that the diameters of all of the apples from his harvest are normally distributed with a mean of 7 cm and a standard deviation of 1.2 cm. He classifies the apples according to their diameters as shown in the following table. Calculate the percentage of small apples in Daniel s harvest. 6

7 7c. [2 marks] Daniel assumes that the diameters of all of the apples from his harvest are normally distributed with a mean of 7 cm and a standard deviation of 1.2 cm. He classifies the apples according to their diameters as shown in the following table. Of the apples harvested, 5% are large apples. Find the value of. 7d. [2 marks] Daniel assumes that the diameters of all of the apples from his harvest are normally distributed with a mean of 7 cm and a standard deviation of 1.2 cm. He classifies the apples according to their diameters as shown in the following table. Find the percentage of medium apples. 7

8 7e. [2 marks] Daniel assumes that the diameters of all of the apples from his harvest are normally distributed with a mean of 7 cm and a standard deviation of 1.2 cm. He classifies the apples according to their diameters as shown in the following table. This year, Daniel estimates that he will grow apples. Estimate the number of large apples that Daniel will grow this year. Printed for Sannarpsgymnasiet International Baccalaureate Organization 2017 International Baccalaureate - Baccalauréat International - Bachillerato Internacional Revision, normal distribution 1a. [3 marks] 8

9 (A1) for normal curve with mean of indicated(a1) for three lines in approximately the correct position(a1) for labels on the three lines (A1)(A1)(A1) 1b. [4 marks] (i) ( ) (A1)(G2) Note: Award for correct region indicated on labelled diagram. (ii) ( ) (A1)(G2) Note: Award for correct region indicated on labelled diagram. 1c. [2 marks] ( ) (A1)(G2) Note: Award for correct region indicated on labelled diagram. 1d. [2 marks] 9

10 Expected number of large size eggs (A1)(G2) 1e. [3 marks] Expected income Note: Award for their correct products, for addition of 4 terms. (A1)(ft)(G3) Note: Follow through from part (b). 2a. [1 mark] (A1) (C1) [1 mark] 2b. [1 mark] (A1) (C1) Note: Accept or. [1 mark] 2c. [2 marks] 10

11 Note: Accept alternative methods. (A1) (C2) [2 marks] 2d. [2 marks] Note: Accept alternative methods. (A1) (C2) [2 marks] 3a. [2 marks] 11

12 (A1)(A1) Notes: Award (A1) for rough sketch of normal curve centred at, (A1) for some indication of as the standard deviation eg, as diagram, or with and shown on the horizontal axis in appropriate places, or for and shown on the horizontal axis in appropriate places. [2 marks] 3b. [1 mark] (A1) Note: Accept only the exact answer. [1 mark] 3c. [2 marks] (G2) Note: Award (G1) for incorrect answer., award (G0) for diagram with correct area shown but [2 marks] 3d. [2 marks] (G2) 12

13 Note: Award (G1) for incorrect answer., award (G0) for diagram with correct area shown but [2 marks] 3e. [2 marks] (R1) Notes: Award (R1) for correct comparison seen. Accept alternative methods, for example, answer to part (c)) used in comparison or a comparison based on scores. (their the Physics result is better (A1)(ft) Notes: Do not award (R0)(A1). Follow through from their answers to part (c) and part (d). [2 marks] 3f. [3 marks] (G3) Notes: Award (G1) for, award (G2) for. Award (G0) for diagram with correct area shown but incorrect answer. [3 marks] 4a. [2 marks] 13

14 Note: Award for correctly shaded area. (A1) (C2) 4b. [2 marks] Note: Award for multiplying by. (A1)(ft) (C2) Note: Follow through from part (a). 4c. [2 marks] 14

15 Note: Award for correctly shaded area. 5a. [2 marks] (A1) (C2) Note: Award for approximate curve with 990 and 1004 in correct place. 5b. [2 marks] (A1) (C2) 15

16 Note: Award for approximate curve with placed to the right of the mean. (A1) (C2) Note: Award full marks only for, or an answer with more than sf resulting from correct rounding of. 5c. [2 marks] 16

17 Note: Award for some indication of symmetry on diagram. OR OR Note: Award for probability with single inequality resulting from symmetry of diagram. 6a. [2 marks] (A1) (C2) (A1) (C2) Notes: Award (A1) for the vertical line labelled as. 17

18 Award for a vertical line drawn to the left of the mean with the area to the left of this line shaded. Accept sd marked on the diagram for (provided line is to the left of the mean). 6b. [4 marks] (i) (A1)(C1) (ii) (A1) Note: Award (A1) for the vertical line labelled as. Award for a vertical line drawn to the right of the mean with the area to the right of this line shaded. Accept 1 sd marked on the diagram for (provided line is to the right of the mean). (A1) (C3) 7a. [3 marks] (i) (G2) Notes: Award for an attempt to use the formula for the mean with a least two rows from the table. (ii) (G1) 7b. [3 marks] (A1) Notes: Award for attempting to use the normal distribution to find the probability or for correct region indicated on labelled diagram. Award (A1) for correct probability. (A1)(ft)(G3) Notes: Award (A1)(ft) for converting their probability into a percentage. 7c. [2 marks] 18

19 Note: Award for attempting to use the normal distribution to find the probability or for correct region indicated on labelled diagram. (A1)(G2) 7d. [2 marks] Note: Award for subtracting their part (b) from 100 or for attempting to use the normal distribution to find the probability region indicated on labelled diagram. or for correct (A1)(ft)(G2) Notes:Follow through from their answer to part (b). Percentage symbol is not required. Accept ( ) if used. 7e. [2 marks] Note: Award for multiplying by (or ). (A1)(G2) Printed for Sannarpsgymnasiet International Baccalaureate Organization 2017 International Baccalaureate - Baccalauréat International - Bachillerato Internacional 19