Question

use the frequency distribution shown below to construct an expanded frequency distribution. high temperatures (°f)
class 20 - 30 31 - 41 42 - 52 53 - 63 64 - 74 75 - 85 86 - 96
frequency, f 17 44 68 69 74 68 25

complete the table below. high temperatures (°f) (round to the nearest hundredth as needed.)
class frequency, f midpoint relative frequency cumulative frequency
20 - 30 17
31 - 41 44
42 - 52 68
53 - 63 69
64 - 74 74
75 - 85 68
86 - 96 25

Explanation:

Step1: Calculate mid - points

The mid - point of a class interval $a - b$ is calculated as $\frac{a + b}{2}$.
For $20 - 30$: $\frac{20+30}{2}=25$
For $31 - 41$: $\frac{31 + 41}{2}=36$
For $42 - 52$: $\frac{42+52}{2}=47$
For $53 - 63$: $\frac{53+63}{2}=58$
For $64 - 74$: $\frac{64+74}{2}=69$
For $75 - 85$: $\frac{75+85}{2}=80$
For $86 - 96$: $\frac{86+96}{2}=91$

Step2: Calculate total frequency

$n=17 + 44+68+69+74+68+25=365$

Step3: Calculate relative frequencies

The relative frequency of a class is $\frac{f}{n}$, where $f$ is the frequency of the class and $n$ is the total frequency.
For $20 - 30$: $\frac{17}{365}\approx0.05$
For $31 - 41$: $\frac{44}{365}\approx0.12$
For $42 - 52$: $\frac{68}{365}\approx0.19$
For $53 - 63$: $\frac{69}{365}\approx0.19$
For $64 - 74$: $\frac{74}{365}\approx0.20$
For $75 - 85$: $\frac{68}{365}\approx0.19$
For $86 - 96$: $\frac{25}{365}\approx0.07$

Step4: Calculate cumulative frequencies

The cumulative frequency of a class is the sum of the frequencies of that class and all previous classes.
For $20 - 30$: $17$
For $31 - 41$: $17+44 = 61$
For $42 - 52$: $61+68 = 129$
For $53 - 63$: $129+69 = 198$
For $64 - 74$: $198+74 = 272$
For $75 - 85$: $272+68 = 340$
For $86 - 96$: $340+25 = 365$

Answer:

Class	Frequency, f	Midpoint	Relative frequency	Cumulative frequency
31 - 41	44	36	0.12	61
42 - 52	68	47	0.19	129
53 - 63	69	58	0.19	198
64 - 74	74	69	0.20	272
75 - 85	68	80	0.19	340
86 - 96	25	91	0.07	365