(a) The distribution of the lives (in days) of 40 transitor batteries is shown in the table: Battery life (in days) 26-30 31-35 36-40 41-45 46-50 51-55 Freq...

First close the gaps between the stated class limits to obtain the true class boundaries, and set out the mid-values, class boundaries and the working columns for the assumed-mean method (assumed mean \(A = 43\)).

Battery life (days)	Class boundaries	Mid-value \(x\)	Frequency \(f\)	\(d = x - 43\)	\(fd\)
26 - 30	25.5 - 30.5	28	4	-15	-60
31 - 35	30.5 - 35.5	33	7	-10	-70
36 - 40	35.5 - 40.5	38	13	-5	-65
41 - 45	40.5 - 45.5	43	8	0	0
46 - 50	45.5 - 50.5	48	6	5	30
51 - 55	50.5 - 55.5	53	2	10	20
Total			40		-145

(a) Histogram

The bars are drawn contiguously on the class boundaries, with frequency on the vertical axis. Because the classes are of equal width (5 days) the bar heights equal the frequencies \(4, 7, 13, 8, 6, 2\).

(b) Mode from the histogram

The tallest bar is the modal class \(36 - 40\) (boundaries \(35.5 - 40.5\)). On the histogram two lines are drawn inside this bar: one from its top-left corner to the top-left corner of the next bar, and one from its top-right corner to the top-right corner of the previous bar. A vertical line dropped from where they cross meets the axis at the mode.

This agrees with the formula, where \(L = 35.5\), \(d_1 = 13 - 7 = 6\), \(d_2 = 13 - 8 = 5\) and \(c = 5\):

\[\text{Mode} = L + \frac{d_1}{d_1 + d_2}\,c = 35.5 + \frac{6}{6 + 5}\times 5 = 35.5 + 2.73 = 38.2\text{ days}\]

(c) Mean (assumed mean \(A = 43\))

\[\bar{x} = A + \frac{\sum fd}{\sum f} = 43 + \frac{-145}{40} = 43 - 3.625 = 39.375 \approx 39.4\text{ days}\]

(d) Percentage living less than 31 days or more than 45 days

Less than 31 days is the class \(26 - 30\): \(4\) batteries. More than 45 days covers the classes \(46 - 50\) and \(51 - 55\): \(6 + 2 = 8\) batteries.

\[\text{Number} = 4 + 6 + 2 = 12\] \[\text{Percentage} = \frac{12}{40}\times 100\% = 30\%\]

Summary: mode \(\approx 38.2\) days, mean \(\approx 39.4\) days, and \(30\%\) of the batteries last under 31 days or over 45 days.