MEI Statistics 1 Data presentation Topic assessment 1. Asha keeps hens so that she has a regular supply of fresh eggs. During March, the numbers of eggs she collected on each day were as follows. 4 5 (i)
5 6
8 4
4 5
6 5
5 6
6 7
7 8
7 8
10 11 18 12 9 13 10 11 14 9
5 6 10
Find the median number of eggs collected.
[1]
(ii) Find the upper quartile, lower quartile and interquartile range.
[3]
(iii) Draw a box and whisker plot for the data.
[3]
(iv) Using your answers to part (ii), identify any outliers.
[3]
(v) Calculate the mean number of eggs laid per day.
[2]
(vi) State the nature of the skewness in your diagram. How do your your answers to parts (i) and (v) support your answer? answer? How do your answers to parts (i) and (ii) support your answer? [3]
2. A random sample of 50 television advertisements was taken and their lengths, x seconds, were recorded. The distribution is illustrated by the histogram below. Frequency per second
1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0
0
10
20
30
40
50
60
70
Time in seconds
(i)
Show that the frequency in the 0 – 0 – 10 10 class interval is 5. Find the remaining remai ning frequencies. Hence illustrate the data by a cumulative cumulative frequency graph. [5]
(ii) Use your cumulative frequency graph to find the 10 th and 90 th percentiles. What proportion of the data lies between these two times? [3] (iii) Calculate estimates of the mean and standard deviation of the length of advertisements.
MEI S1 Data presentation Assessment It was later found that the times recorded within each interval were in fact at the upper class boundary for that interval. (iv) State, with a reason, how this information would affect your answers to part (iii). [2]
3. The cumulative frequency graph below illustrates the distribution of the weights, x kg, of 150 newly- born babies in a hospital’s maternity unit. Cumulative frequency
150
150 130
142
125 107
100 75
74 52
50 25
22 7
0 2.5 2.75 3.0 3.25 3.5 3.75 4.0 4.25 4.5 Weight in kg (i)
Use the cumulative frequency graph to estimate the median and interquartile range. [3]
(ii) Draw a box and whisker plot for the data.
[3]
(iii) Copy and complete the following frequency table. Hence calculate an estimate of the mean weight of the 150 babies.
[3]
Weight of baby (kg) 2.50 x 2.75 2.75 x 3.00
Frequency 7 15
3.00 x 3.25 3.25 x 3.50 3.50 x 3.75 3.75 x 4.00 4.00 x 4.25 4.25 x 4.50
During the same period of time, 18 babies were born in the special care baby unit of the hospital. The mean weight of all 168 babies was found to be 3.35 kg (to 3 significant figures).
MEI S1 Data presentation Assessment (iv) Calculate an estimate of the mean weight of the 18 babies in the special care unit. [3] (v) What effect will the addition of the 18 extra data items have on the median and on the interquartile range? Explain your reasoning. [3]
4. A motoring magazine carried out a survey of the value of petrol-driven cars that were five years old. In the survey, the value of each car was expressed as a percentage of its value when new. The results of the survey are summarised in the following table. Percentage of original value ( x)
Number of cars
15 x 20
4 12 18 13 6 5 2
20 x 25 25 x 30 30 x 35 35 x 40 40 x 45 45 x 55
(i)
Draw a histogram on graph paper to illustrate the data.
[4]
(ii) Calculate an estimate of the median of the data.
[2]
(iii) Calculate estimates of the mean and standard deviation of the data, giving your answers correct to 2 decimal places. Hence identify any outliers, explaining your method. [7] A similar survey of 60 diesel driven ca rs produced a mean of 34.3% and a standard deviation of 11.7%. (iv) Use these statistics to compare the values of petrol and diesel cars five years after they were purchased as new. [2]
