⑦ Matches are sold in boxes which have 7 printed on them, 'Average contents 48 matches: 100 boxes were picked at random and their contents were counted. The results are given in Table 14.11. matches per box 39-41 42-44 45-47 48-50 51-53 54-56 frequency 3 13 26 38 15 5 a Draw a histogram for this distribution. b By considering the modal class, mode and mean of the data, decide whether or not the statement on the boxes is true. c State the median class. d Estimate the median using the formulae?

Qawsw

2 years ago

a) Here is the histogram for the distribution:

[Insert histogram image]

b) The modal class is 48-50, with a frequency of 38. The mode is the value that appears most frequently, which is 49 (since 48-50 is the modal class). The mean is calculated as:

Mean = (Sum of frequencies x Midpoint of each class) / Total frequency
= (3 x 40 + 13 x 43 + 26 x 46 + 38 x 49 + 15 x 52 + 5 x 55) / 100
= 4840 / 100
= 48.4

Since the mean (48.4) is close to the modal value (49), and the statement on the boxes is "Average contents 48 matches", we can conclude that the statement is approximately true.

c) The median class is the class that contains the middle value of the data. Since there are 100 data points, the middle value is the 50th value. By counting the frequencies, we find that the 50th value falls in the class 48-50. Therefore, the median class is 48-50.

d) To estimate the median using the formula, we need to find the median class and the cumulative frequency up to the median class.

Median class = 48-50
Cumulative frequency up to median class = 3 + 13 + 26 + 38 = 80

Using the formula:

Median = Lower limit of median class + ((Cumulative frequency / 2) - Frequency of preceding class) x Class width
= 48 + ((80 / 2) - 38) x 2
= 48 + 1 x 2
= 50

So, the estimated median is 50.