descriptive statistics and Cumulative_frequency hw

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Cumulative frequency [160 marks]

1a.

Adam is a beekeeper who collected data about monthly honey production in his bee hives. The data for six of his hives is shown in the following table.

The relationship between the variables is modelled by the regression line with equation .

Write down the value of and of .

P = aN + b

a b

1b. Use this regression line to estimate the monthly honey production from a hive that has 270 bees.

1c.

Adam has 200 hives in total. He collects data on the monthly honey production of all the hives. This data is shown in the following cumulative frequency graph.

Adam’s hives are labelled as low, regular or high production, as defined in the following table.

Write down the number of low production hives.

1d.

Adam knows that 128 of his hives have a regular production.

Find the value of ;k

1e. Find the number of hives that have a high production.

1f. Adam decides to increase the number of bees in each low production hive. Research suggests that there is a probability of 0.75 that a low production hive becomes a regular production hive. Calculate the probability that 30 low production hives become regular production hives.

[3 marks]

[2 marks]

[1 mark]

[3 marks]

[2 marks]

[3 marks]

2a.

A city hired 160 employees to work at a festival. The following cumulative frequency curve shows the number of hours employees worked during the festival.

Find the median number of hours worked by the employees.

2b. Write down the number of employees who worked 50 hours or less.

2c.

The city paid each of the employees £8 per hour for the first 40 hours worked, and £10 per hour for each hour they worked after the first 40 hours.

Find the amount of money an employee earned for working 40 hours;

2d. Find the amount of money an employee earned for working 43 hours.

2e. Find the number of employees who earned £200 or less.

2f. Only 10 employees earned more than £ . Find the value of .k k

3a.

Ten students were surveyed about the number of hours, , they spent browsing the Internet during week 1 of the school year. The results of the survey are given below.

Find the mean number of hours spent browsing the Internet.

10∑ i=1 xi = 252, σ = 5 and median = 27.

[2 marks]

[1 mark]

[3 marks]

[4 marks]

[2 marks]

3b. 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.

3c. 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.

3d.

During week 4, the survey was extended to all 200 students in the school. The results are shown in the cumulative frequency graph:

(i) Find the number of students who spent between 25 and 30 hours browsing the Internet.

(ii) Given that 10% of the students spent more than k hours browsing the Internet, find the maximum value of .k

[2 marks]

[6 marks]

4a.

A school collects cans for recycling to raise money. Sam’s class has 20 students.

The number of cans collected by each student in Sam’s class is shown in the following stem and leaf diagram.

Find the median number of cans collected.

4b.

The following box-and-whisker plot also displays the number of cans collected by students in Sam’s class.

(i) Write down the value of .

(ii) The interquartile range is 14. Find the value of .

4c. Sam’s class collected 745 cans. They want an average of 40 cans per student.

How many more cans need to be collected to achieve this target?

4d.

There are 80 students in the school.

The students raise $0.10 for each recycled can.

(i) Find the largest amount raised by a student in Sam’s class.

(ii) The following cumulative frequency curve shows the amounts in dollars raised by all the students in the school. Find the percentage of students in the school who raised more money than anyone in Sam’s class.

[2 marks]

[3 marks]

[5 marks]

4e.

The mean number of cans collected is 39.4. The standard deviation is 18.5.

Each student then collects 2 more cans.

(i) Write down the new mean.

(ii) Write down the new standard deviation.

[2 marks]

5a.

The following cumulative frequency graph shows the monthly income, dollars, of families.

Find the median monthly income.

I 2000

5b. (i) Write down the number of families who have a monthly income of dollars or less.

(ii) Find the number of families who have a monthly income of more than dollars.

2000

4000

5c. The families live in two different types of housing. The following table gives information about the number of families living in each type of housing and their monthly income .

Find the value of .

2000 I

5d. A family is chosen at random.

(i) Find the probability that this family lives in an apartment.

(ii) Find the probability that this family lives in an apartment, given that its monthly income is greater than dollars.4000

5e. Estimate the mean monthly income for families living in a villa.

[2 marks]

[4 marks]

[2 marks]

6a.

The weights in grams of 80 rats are shown in the following cumulative frequency diagram.

Do NOT write solutions on this page.

Write down the median weight of the rats.

6b. Find the percentage of rats that weigh 70 grams or less.

6c. The same data is presented in the following table.

Weights grams

Frequency

Write down the value of .

0 ⩽ w ⩽ 30 30 < w ⩽ 60 60 < w ⩽ 90 90 < w ⩽ 120 p

45 q

[1 mark]

[3 marks]

[2 marks]

6d. The same data is presented in the following table.

Weights grams

Frequency

Find the value of .

0 ⩽ w ⩽ 30 30 < w ⩽ 60 60 < w ⩽ 90 90 < w ⩽ 120 p

45 q

6e. The same data is presented in the following table.

Weights grams

Frequency

Use the values from the table to estimate the mean and standard deviation of the weights.

0 ⩽ w ⩽ 30 30 < w ⩽ 60 60 < w ⩽ 90 90 < w ⩽ 120 p

45 q

6f. Assume that the weights of these rats are normally distributed with the mean and standard deviation estimated in part (c).

Find the percentage of rats that weigh 70 grams or less.

6g. Assume that the weights of these rats are normally distributed with the mean and standard deviation estimated in part (c).

A sample of five rats is chosen at random. Find the probability that at most three rats weigh 70 grams or less.

[2 marks]

[3 marks]

[2 marks]

[3 marks]

7a.

The following is a cumulative frequency diagram for the time t, in minutes, taken by 80 students to complete a task.

Find the number of students who completed the task in less than 45 minutes.

7b. Find the number of students who took between 35 and 45 minutes to complete the task.

7c. Given that 50 students take less than k minutes to complete the task, find the value of .k

8a.

A running club organizes a race to select girls to represent the club in a competition.

The times taken by the group of girls to complete the race are shown in the table below.

Find the value of and of .

[2 marks]

[3 marks]

[2 marks]

[4 marks]

8b. A girl is chosen at random.

(i) Find the probability that the time she takes is less than minutes.

(ii) Find the probability that the time she takes is at least minutes.

8c. A girl is selected for the competition if she takes less than minutes to complete the race.

Given that of the girls are not selected,

(i) find the number of girls who are not selected;

(ii) find .

40%

8d. Girls who are not selected, but took less than minutes to complete the race, are allowed another chance to be selected. The new times taken by these girls are shown in the

cumulative frequency diagram below.

(i) Write down the number of girls who were allowed another chance.

(ii) Find the percentage of the whole group who were selected.

[3 marks]

[4 marks]

9a.

The cumulative frequency curve below represents the marks obtained by 100 students.

Find the median mark.

9b. Find the interquartile range.

10a.

The cumulative frequency curve below represents the heights of 200 sixteen-year-old boys.

Use the graph to answer the following.

Write down the median value.

10b. A boy is chosen at random. Find the probability that he is shorter than .161 cm

[2 marks]

[3 marks]

[1 mark]

[2 marks]

10c. Given that of the boys are taller than , find h .

82% h cm

11a.

A scientist has 100 female fish and 100 male fish. She measures their lengths to the nearest cm. These are shown in the following box and whisker diagrams.

Find the range of the lengths of all 200 fish.

11b. Four cumulative frequency graphs are shown below.

Which graph is the best representation of the lengths of the female fish?

12a.

The following table gives the examination grades for 120 students.

Find the value of

(i) p ;

(ii) q .

12b. Find the mean grade.

[3 marks]

[2 marks]

[4 marks]

[2 marks]

12c. Write down the standard deviation.

13a.

A fisherman catches 200 fish to sell. He measures the lengths, l cm of these fish, and the results are shown in the frequency table below.

Calculate an estimate for the standard deviation of the lengths of the fish.

13b. A cumulative frequency diagram is given below for the lengths of the fish.

Use the graph to answer the following.

(i) Estimate the interquartile range.

(ii) Given that of the fish have a length more than , find the value of k.

40% k cm

13c. In order to sell the fish, the fisherman classifies them as small, medium or large.

Small fish have a length less than .

Medium fish have a length greater than or equal to but less than .

Large fish have a length greater than or equal to .

Write down the probability that a fish is small.

20 cm

20 cm 60 cm

60 cm

[1 mark]

[3 marks]

[6 marks]

[2 marks]

13d. The cost of a small fish is , a medium fish , and a large fish .

Copy and complete the following table, which gives a probability distribution for the cost .

$4 $10 $12

13e. Find .E(X)

14a.

The following is a cumulative frequency diagram for the time t, in minutes, taken by 80 students to complete a task.

Write down the median.

14b. Find the interquartile range.

14c. Complete the frequency table below.

[2 marks]

[1 mark]

[3 marks]

[2 marks]

Printed for American Creativity Academy

International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®

15a.

In a school with 125 girls, each student is tested to see how many sit-up exercises (sit-ups) she can do in one minute. The results are given in the table below.

(i) Write down the value of p.

(ii) Find the value of q.

15b. Find the median number of sit-ups.

15c. Find the mean number of sit-ups.

[3 marks]

[2 marks]

Cumulative frequency [160 marks]