Any necessary formulae are given on the formula sheet provided on DUO.
1. Each member of a group of 35 people went walking on a weekend.
The time is taken for the walk, π₯ minutes, for each member of the group was recorded. These times are summarised in the following box and whisker plot.
(a) Find the range of the times.
(b) Find the interquartile range of the times.
For these 35 people ∑ π₯ = 3850 and ∑ π₯2 = 575,960
(c) Calculate the mean time taken on the walk,
(d) Calculate the standard deviation of the times taken on the walk.
Louise defines an outlier as a value more than 3 standard deviations above or below the mean.
(e) State how many outliers Louise would say there are in these data, giving a reason for your answer.
2. The table gives information about the numbers of students from different types of schools who applied to Cambridge University in 2016:
|
Type of school |
Gender |
Total |
|
|
Male |
Female |
||
|
Maintained |
3674 |
2899 |
6573 |
|
Independent |
1510 |
1268 |
2778 |
|
Other and overseas |
300 |
312 |
612 |
|
Total |
5484 |
4479 |
9963 |
Richard is going to take a sample of 200 of these students stratified by gender.
(a) Work out how many female students there should be in this sample.
(b) Describe a situation when it would not be appropriate to take a sample stratified by gender.
Richard could have used a different category for his stratified sample.
(c) What is this different category?
A student is to be chosen at random from the 9963 students. F is the event that the student chosen is female.
I am the event that the student chosen is from an independent school. M is the event that the student chosen is from a maintained school.
(d) Explain why event F and event I are not mutually exclusive.
(e) Find the probability of I or M.
(f) Find the probability of F given M.
3. The table shows information about the amount of time that each member of a group of 46 teenagers spent on social media during one day.
|
Time spent on social media (t minutes) |
Frequency |
|
0 ≤ t < 50 |
1 |
|
50 ≤ t < 100 |
4 |
|
100 ≤ t < 150 |
8 |
|
150 ≤ t <200 |
17 |
|
200 ≤ t < 300 |
16 |
(a) (i) Use linear interpolation to find an estimate of the median time spent on social media by 46 teenagers. The average person will spend approximately 116 minutes on social media each day.(Source: www.socialmediatoday.com)
(ii) Compare the amount of time spent on social media by the 46 teenagers with the amount of time spent on social media by the average person.
The table below gives the mean, the standard deviation and the median for the times spent on social media during one day by a sample of sixty-year-olds.
|
Mean (minutes) |
Standard deviation (minutes) |
Median (minutes) |
|
125 |
25 |
130 |
£1000, was recorded for each manager.
The director then calculated a performance score, π₯, for each of these managers. The results are summarised in the following statistics.
∑ π₯ = 465 ∑ π¦ = 562 ππ₯π₯ = 2492 ∑ π¦2 = 23 140 ∑ π₯π¦ = 19428
(a) (i) Show that ππ₯π¦ = 2006
(ii) Find ππ¦π¦
(b) Find the product moment correlation coefficient between performance score and annual salary.
(c) Describe the correlation between performance score and annual salary.
The director believes that there is a linear relationship between performance score and annual salary.
(d) Calculate the equation of the regression line of π¦ on π₯, in the form π¦ = π + ππ₯. Give the value of π and the value of π to 3 significant figures.
(e) Give an interpretation of the value of π.
5. A geography student is investigating the relationship between the size of a shopping center (measured by the number of shops it contains) and the mean distance traveled by shoppers to reach the shopping center.
She obtains the following data:
|
Centre |
A |
B |
C |
D |
E |
F |
|
No. of shops |
6 |
20 |
15 |
30 |
30 |
60 |
|
Mean distance (km) |
0.5 |
2.1 |
2.4 |
4.1 |
3.9 |
6.2 |
Calculate Spearman’s rank the correlation coefficient between mean distance traveled and number of shops
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