Date | May 2018 | Marks available | 2 | Reference code | 18M.1.AHL.TZ2.H_3 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 2 |
Command term | Find | Question number | H_3 | Adapted from | N/A |
Question
The discrete random variable X has the following probability distribution, where p is a constant.
Find the value of p.
Find μ, the expected value of X.
Find P(X > μ).
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
equating sum of probabilities to 1 (p + 0.5 − p + 0.25 + 0.125 + p3 = 1) M1
p3 = 0.125 =
p= 0.5 A1
[2 marks]
μ = 0 × 0.5 + 1 × 0 + 2 × 0.25 + 3 × 0.125 + 4 × 0.125 M1
= 1.375 A1
[2 marks]
P(X > μ) = P(X = 2) + P(X = 3) + P(X = 4) (M1)
= 0.5 A1
Note: Do not award follow through A marks in (b)(i) from an incorrect value of p.
Note: Award M marks in both (b)(i) and (b)(ii) provided no negative probabilities, and provided a numerical value for μ has been found.
[2 marks]
Examiners report
Syllabus sections
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22M.1.SL.TZ1.3b.ii:
Find the mean age of all the adults surveyed.
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22M.2.SL.TZ1.2b:
Find the standard deviation.
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19M.2.AHL.TZ1.H_3a.i:
Find the mean.
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19M.2.AHL.TZ1.H_3a.ii:
Find the standard deviation.
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19M.2.AHL.TZ1.H_3b.i:
the mean.
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19M.2.AHL.TZ1.H_3b.ii:
the standard deviation.
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19M.1.SL.TZ2.T_6b.ii:
Write down the approximate number of snacks whose amount of sugar ranges from 18 to 20 grams.
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19M.2.SL.TZ1.S_1c:
Find the interquartile range.
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19M.1.SL.TZ1.T_2b:
Write down the median speed for these animals.
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18M.2.SL.TZ1.T_2a:
State the alternative hypothesis.
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18M.2.SL.TZ1.T_2d.i:
Write down the χ2 statistic.
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18M.2.SL.TZ1.T_2d.ii:
Write down the associated p-value.
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18N.1.SL.TZ0.T_2b:
Find the median length of the rods.
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18N.1.SL.TZ0.T_2c.ii:
Calculate the interquartile range.
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18M.1.SL.TZ1.T_6c.i:
Write down the mid-interval value for the 100 < x ≤ 150 group.
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18M.1.SL.TZ2.T_12a:
Write down the mid-interval value for 10 ≤ t < 15.
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18M.1.SL.TZ2.T_12b.i:
Write down the total number of customers in terms of k.
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18N.2.SL.TZ0.T_3b:
Find the final examination result required to obtain the highest possible grade.
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16N.2.SL.TZ0.T_1h:
Write down a reason why this estimate is not reliable.
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19M.2.SL.TZ1.T_3b:
Write down the modal number of pets.
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19M.2.SL.TZ1.T_3c.i:
For these data, write down the median number of pets.
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19M.2.SL.TZ1.T_3e.i:
State the null hypothesis.
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19M.1.SL.TZ2.T_14c:
To stimulate reasonable pricing, the city offers a free permit to the sellers whose price of a kilogram of tomatoes is in the lowest 20 %.
Find the highest price that a seller can charge and still receive a free permit.
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19M.1.SL.TZ2.T_14b:
Find the probability that the price of a kilogram of tomatoes, chosen at random, will be between 2.00 and 3.00 euro.
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18N.2.SL.TZ0.T_3a.i:
Find the median of the examination results.
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18N.1.SL.TZ0.T_2c.i:
Calculate the lower quartile.
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19M.1.SL.TZ1.T_2a:
State whether speed is a continuous or discrete variable.
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19M.2.SL.TZ1.T_3a:
Write down the total number of people, from this group, who are pet owners.
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18M.1.SL.TZ1.T_2b.i:
Find the standard deviation
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19M.1.SL.TZ2.T_6b.i:
Write down the interquartile range for this data.
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18M.1.SL.TZ1.T_6b:
Complete the table.
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18M.1.SL.TZ1.S_2a:
Find the value of the interquartile range.
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19M.1.SL.TZ2.S_8b:
Find the value of the interquartile range.
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17M.1.SL.TZ2.S_8a.ii:
Write down the number of employees who worked 50 hours or less.
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19M.1.SL.TZ1.T_2d.ii:
For these eight animals write down the standard deviation.
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19M.1.SL.TZ2.S_8a:
The range of the data is 16. Find the value of .
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18M.1.SL.TZ2.T_12c:
Hence, complete the histogram.
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18M.2.SL.TZ2.T_2b.i:
Use the cumulative frequency curve to find the median distance.
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18M.2.SL.TZ2.T_2b.iii:
Use the cumulative frequency curve to find the upper quartile.
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18M.2.SL.TZ2.T_2d:
Write down the percentage of buses that travelled a distance greater than the upper quartile.
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18M.2.SL.TZ2.T_2e:
Find the number of buses that travelled a distance less than or equal to 12 000 km.
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18M.2.SL.TZ2.T_2g:
The smallest distance travelled by one of the buses was 2500 km.
The longest distance travelled by one of the buses was 23 000 km.On graph paper, draw a box-and-whisker diagram for these data. Use a scale of 2 cm to represent 5000 km.
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17M.2.SL.TZ1.T_5b:
Find the median test grade of the students.
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17M.2.SL.TZ1.S_1b.i:
Find the mean.
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17N.2.SL.TZ0.T_1f.i:
Write down the -value for the test;
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18M.1.SL.TZ1.T_6a:
Write down the median.
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18N.2.SL.TZ0.T_3d.i:
Calculate an estimate of the mean examination result.
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17M.1.SL.TZ2.S_8a.i:
Find the median number of hours worked by the employees.
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19M.2.SL.TZ1.T_3c.iii:
For these data, write down the upper quartile.
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19M.1.SL.TZ2.T_6c:
The health inspector visits two school cafeterias. She inspects the same number of meals at each cafeteria. The data is shown in the following box-and-whisker diagrams.
Meals prepared in the school cafeterias are required to have less than 10 grams of sugar.
State, giving a reason, which school cafeteria has more meals that do not meet the requirement.
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18M.1.SL.TZ2.S_3b.i:
Write down the value of the new mean.
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17M.1.SL.TZ2.S_8b.i:
Find the amount of money an employee earned for working 40 hours;
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19M.2.SL.TZ1.T_3c.ii:
For these data, write down the lower quartile.
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18M.2.SL.TZ2.T_2b.ii:
Use the cumulative frequency curve to find the lower quartile.
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17M.1.SL.TZ2.S_8b.ii:
Find the amount of money an employee earned for working 43 hours.
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18N.2.SL.TZ0.T_3d.ii:
Calculate an estimate of the standard deviation, giving your answer correct to three decimal places.
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18M.2.SL.TZ2.T_2f:
Find the value of m.
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18N.1.SL.TZ0.T_2a:
Write down the modal length of the rods.
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17M.1.SL.TZ2.S_8d:
Only 10 employees earned more than £. Find the value of .
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20N.1.SL.TZ0.T_3b:
Use your graphic display calculator to find an estimate of the standard deviation of the weights of mangoes from this harvest.
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19M.1.SL.TZ2.S_8c:
Find the mean number of hours that the girls in this group spent watching television that week.
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18M.2.SL.TZ1.T_2g:
Given that this flight was not heavily delayed, find the probability that it travelled between 500 km and 5000 km.
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19M.1.SL.TZ1.T_2c:
Write down the range of the animal speeds.
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19M.2.SL.TZ1.S_1b:
Find the value of .
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17N.2.SL.TZ0.T_1b.i:
Write down, for , the modal class;
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17M.2.AHL.TZ2.H_1b:
Calculate the mean score.
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18M.2.SL.TZ2.T_2c:
Hence write down the interquartile range.
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22M.1.SL.TZ1.3b.i:
One of the adults surveyed is years old. Estimate the age of their eldest child.
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17M.2.AHL.TZ2.H_1a:
One of the players is chosen at random. Find the probability that this player’s score was 5 or more.
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19M.1.SL.TZ2.S_8d.i:
Find the total number of hours the group of boys spent watching television that week.
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17M.2.SL.TZ1.S_1a.ii:
Find the value of the range.
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20N.1.SL.TZ0.S_8d:
Find .
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17M.1.SL.TZ2.S_8c:
Find the number of employees who earned £200 or less.
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20N.1.SL.TZ0.S_8b:
Write down the value of the median distance in kilometres (km).
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20N.2.SL.TZ0.S_9c:
Find the probability that the bus journey takes less than minutes.
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20N.2.SL.TZ0.S_9d:
Find the probability that Fiona will arrive on time.
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20N.2.SL.TZ0.S_9e:
This year, Fiona will go to school on days.
Calculate the number of days Fiona is expected to arrive on time.
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20N.1.SL.TZ0.T_7d.ii:
Using your answers to parts (b) and (c), find the height of Flower .
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19M.1.SL.TZ2.S_8e.i:
the mean number of hours that the group of boys spent watching television.
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17M.2.SL.TZ1.S_1a.i:
Write down the mode.
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18M.1.SL.TZ1.S_2b:
One student sent k text messages, where k > 11 . Given that k is an outlier, find the least value of k.
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20N.2.SL.TZ0.S_9b:
Find .
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17M.1.SL.TZ2.T_7c:
Find the value of and of .
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18M.1.AHL.TZ2.H_3b.i:
Find μ, the expected value of X.
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20N.1.SL.TZ0.S_8c:
Find the value of .
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20N.1.SL.TZ0.T_7b:
Using this information, write down an equation in and .
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20N.1.SL.TZ0.T_7c:
Write down a second equation in and .
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18M.1.AHL.TZ2.H_3b.ii:
Find P(X > μ).
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16N.2.SL.TZ0.S_8c:
During week 3 each student spent 5% less time browsing the Internet than during week 1. For week 3, find
(i) the median;
(ii) the variance.
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20N.2.SL.TZ0.S_9a:
Find the probability that it will take Fiona between minutes and minutes to walk to the bus stop.
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19M.2.AHL.TZ1.H_3c:
A ninth student also takes the test.
Explain why the median is unchanged.
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EXN.2.SL.TZ0.1:
A data set consisting of test scores has mean . One test score of requires a second marking and is removed from the data set.
Find the mean of the remaining test scores.
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22M.2.SL.TZ1.2a:
Find the value of .
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19M.1.SL.TZ2.T_14a.ii:
Find the price that is two standard deviations above the mean price.
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19M.1.SL.TZ1.T_2d.i:
For these eight animals find the mean speed.
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17N.1.SL.TZ0.T_1a.ii:
For the students in this group write down the median age.
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19M.2.SL.TZ1.T_3f:
Write down the number of degrees of freedom for this test.
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16N.1.SL.TZ0.T_2b:
Draw a box-and-whisker diagram on the grid below to represent the Vitamin C content, in milligrams, for this sample.
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19M.1.SL.TZ2.T_6a:
State what 13 represents in the given diagram.
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17N.2.SL.TZ0.T_1c.i:
Use your graphic display calculator to estimate the mean of ;
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18M.2.SL.TZ1.T_2e:
State, with a reason, whether you would reject the null hypothesis.
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16N.2.SL.TZ0.T_1c:
Plot the point on your scatter diagram and label this point M.
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17M.1.SL.TZ2.T_7b:
Using the mean score, write down a second equation in terms of and .
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20N.1.SL.TZ0.T_3c:
On the grid below, draw a histogram for the data in the table.
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16N.2.SL.TZ0.T_1e:
Write down the equation of the regression line on for these eight male students.
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17N.1.SL.TZ0.T_1b:
Draw a box-and-whisker diagram, for these students’ ages, on the following grid.
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17M.1.SL.TZ2.T_7a:
Write down an equation, in terms of and , for the total number of times the die was rolled.
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18M.1.SL.TZ2.S_3a:
Find n.
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19M.2.SL.TZ1.S_1a:
For these data, find the mean distance from a student’s home to school.
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17M.2.SL.TZ1.S_1b.ii:
Find the variance.
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19M.1.SL.TZ2.S_8d.ii:
Find the mean number of hours that all 30 girls and boys spent watching television that week.
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18M.2.SL.TZ1.T_2h:
Two flights are chosen at random from those which were slightly delayed.
Find the probability that each of these flights travelled at least 5000 km.
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17N.2.SL.TZ0.T_1g:
State the result of the test. Give a reason for your answer.
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16N.2.SL.TZ0.S_8b:
During week 2, the students worked on a major project and they each spent an additional five hours browsing the Internet. For week 2, write down
(i) the mean;
(ii) the standard deviation.
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18M.1.SL.TZ1.T_6c.ii:
Using the table, calculate an estimate for the mean number of people being followed on the social media website by these 160 students.
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18M.1.SL.TZ2.S_3b.ii:
Find the value of the new variance.
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20N.1.SL.TZ0.S_8a:
Find the value of .
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20N.1.SL.TZ0.S_8e:
The first athletes that completed the race won a prize.
Given that an athlete took between and minutes to complete the race, calculate the probability that they won a prize.
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18M.2.SL.TZ1.T_2b:
Calculate the expected frequency of flights travelling at most 500 km and arriving slightly delayed.
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16N.2.SL.TZ0.S_8a:
Find the mean number of hours spent browsing the Internet.
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18M.1.SL.TZ1.T_2b.ii:
Find the interquartile range.
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18M.1.SL.TZ2.T_12b.ii:
Calculate the value of k.
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17N.2.SL.TZ0.T_1b.ii:
Write down, for , the mid-interval value of the modal class.
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19M.2.SL.TZ1.T_3e.ii:
State the alternative hypothesis.
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18M.1.SL.TZ1.T_2a:
Find the value of x.
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18M.2.SL.TZ1.T_2c:
Write down the number of degrees of freedom.
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17M.2.SL.TZ1.T_5f.ii:
Calculate the expected number of students that spent at least 90 minutes preparing for the test.
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17N.2.SL.TZ0.T_1a:
State whether is a discrete or a continuous variable.
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17N.2.SL.TZ0.T_1f.ii:
Write down the statistic.
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16N.2.SL.TZ0.T_1d:
Write down the value of , the Pearson’s product–moment correlation coefficient, for these data.
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19M.2.SL.TZ1.T_3d:
Write down the ratio of teenagers to non-teenagers in its simplest form.
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17N.1.SL.TZ0.T_1a.i:
For the students in this group find the mean age;
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20N.1.SL.TZ0.T_7d.i:
Using your answers to parts (b) and (c), find the height of Flower .
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19M.2.SL.TZ1.T_3g:
Calculate the expected number of teenagers that prefer cats.
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17M.2.SL.TZ1.T_5a.i:
Calculate the mean test grade of the students;
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20N.1.SL.TZ0.T_7a:
Find the height of Flower null.
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16N.1.SL.TZ0.T_2a:
Giving your answer to one decimal place, write down the value of
(i) the median level of Vitamin C content of the oranges in the sample;
(ii) the lower quartile;
(iii) the upper quartile.
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17M.2.SL.TZ1.T_5e:
Given that the first student chosen at random scored a grade 5 or higher, find the probability that both students scored a grade 6.
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20N.1.SL.TZ0.T_3a:
Write down the modal group for these data.
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17M.2.SL.TZ1.T_5d:
Find the probability that this student scored a grade 5 or higher.
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18M.2.SL.TZ1.T_2f:
Write down the probability that this flight arrived on time.
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17N.2.SL.TZ0.T_1c.ii:
Use your graphic display calculator to estimate the standard deviation of .
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17N.2.SL.TZ0.T_1d:
Find the expected frequency of students choosing the Science category and obtaining 31 to 40 correct answers.
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17N.2.SL.TZ0.T_1e.i:
Write down the null hypothesis for this test;
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17N.2.SL.TZ0.T_1e.ii:
Write down the number of degrees of freedom.
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19M.2.SL.TZ1.T_3h:
Use your graphic display calculator to find the -value for this test.
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18N.2.SL.TZ0.T_3a.ii:
Find the interquartile range.
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18N.2.SL.TZ0.T_3c.i:
Write down the modal class.
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18N.2.SL.TZ0.T_3c.ii:
Write down the mid-interval value of the modal class.
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18N.2.SL.TZ0.T_3e:
The teacher sets a grade boundary that is one standard deviation below the mean.
Use the cumulative frequency graph to estimate the number of students whose final examination result was below this grade boundary.
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18M.2.SL.TZ2.T_2a:
Find the number of buses that travelled a distance between 15000 and 20000 kilometres.
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16N.2.SL.TZ0.T_1a:
On graph paper, draw a scatter diagram for these data. Use a scale of 2 cm to represent 5 hours on the -axis and 2 cm to represent 10 points on the -axis.
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16N.2.SL.TZ0.T_1b:
(i) , the mean number of hours spent on social media;
(ii) , the mean number of IB Diploma points.
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16N.2.SL.TZ0.T_1f:
Draw the regression line, from part (e), on your scatter diagram.
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16N.2.SL.TZ0.T_1g:
Use the given equation of the regression line to estimate the number of IB Diploma points that this girl obtained.
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17M.2.SL.TZ1.T_5a.ii:
Calculate the standard deviation.
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17M.2.SL.TZ1.T_5c:
Find the interquartile range.
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17M.2.SL.TZ1.T_5f.i:
Calculate the probability that a student chosen at random spent at least 90 minutes preparing for the test.
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19M.2.SL.TZ1.T_3i:
State the conclusion for this test. Give a reason for your answer.