Question

1. you are interested in how much time you spend on instagram™ so you recorded the number of minutes spent browsing your newsfeed each day for three weeks. you obtain the following data:

35 62 25 29 37 27 64
17 24 46 14 29 28 54
17 39 32 39 73 41 23
a. create a grouped frequency distribution table that (i) has the best possible class interval width and (ii) an appropriate number of class intervals.
b.

Explanation:

Step1: Find the range

First, find the minimum and maximum values in the data set. The minimum value ($\min$) is 14, and the maximum value ($\max$) is 73. The range ($R$) is calculated as $R=\max - \min$.
$R = 73 - 14 = 59$

Step2: Determine the number of classes

A common rule is to use the "2 to the k rule", where $2^k \geq n$, and $n$ is the number of data points. Here, $n = 21$ (since there are 3 weeks $\times$ 7 days = 21 days). Let's check: $2^4 = 16 < 21$, $2^5 = 32 \geq 21$, so we can use $k = 5$ or a bit more. Alternatively, we can use the formula for class width ($w$): $w=\frac{R}{k}$, where $k$ is the number of classes. Let's aim for 5 - 7 classes. Let's try $k = 6$. Then $w=\frac{59}{6}\approx 10$ (we round up to a convenient number, 10 is a good choice here).

Step3: Define the class intervals

Start with the minimum value, 14. The first class interval: 10 - 19 (since 14 is in this range, and width 10). Then the next classes: 20 - 29, 30 - 39, 40 - 49, 50 - 59, 60 - 69, 70 - 79. Let's check if all data points are covered.

Step4: Count the frequency for each class

• Class 10 - 19: Data points are 17, 14, 17. So frequency ($f$) = 3.
• Class 20 - 29: Data points are 25, 29, 27, 24, 29, 28, 23. So $f = 7$.
• Class 30 - 39: Data points are 35, 37, 39, 32, 39. So $f = 5$.
• Class 40 - 49: Data points are 46, 41. So $f = 2$.
• Class 50 - 59: Data point is 54. So $f = 1$.
• Class 60 - 69: Data points are 62, 64. So $f = 2$.
• Class 70 - 79: Data point is 73. So $f = 1$. Wait, but we initially thought of 6 classes, but with width 10, starting at 10, we have 7 classes. Let's check the range again. 73 - 14 = 59. If we use width 10, the number of classes is $\lceil\frac{59}{10}

ceil = 6$? Wait, 10*6 = 60, 14 + 60 = 74, which covers 73. Wait, maybe my initial class start was wrong. Let's start at 14? No, class intervals are usually with lower limits as multiples of the width or rounded. Let's correct:

Alternative approach: Let's use class width 10, and start at 10. So classes: 10 - 19, 20 - 29, 30 - 39, 40 - 49, 50 - 59, 60 - 69, 70 - 79. But let's count again:

Data set: 35, 62, 25, 29, 37, 27, 64, 17, 24, 46, 14, 29, 28, 54, 17, 39, 32, 39, 73, 41, 23.

10 - 19: 14, 17, 17 → 3

20 - 29: 23, 24, 25, 27, 28, 29, 29 → 7 (wait, 23 is 20 - 29, yes. So 23,24,25,27,28,29,29: 7 values)

30 - 39: 32, 35, 37, 39, 39 → 5 (32,35,37,39,39: 5)

40 - 49: 41, 46 → 2 (41,46: 2)

50 - 59: 54 → 1 (54:1)

60 - 69: 62, 64 → 2 (62,64:2)

70 - 79: 73 → 1 (73:1)

Now, let's check the total frequency: 3 + 7 + 5 + 2 + 1 + 2 + 1 = 21, which matches the number of data points.

Step5: Create the grouped frequency distribution table

Class Interval	Frequency ($f$)
20 - 29	7
30 - 39	5
40 - 49	2
50 - 59	1
60 - 69	2
70 - 79	1

We can also calculate relative frequency, cumulative frequency, etc., but the question asks for grouped frequency distribution table with best class interval width and appropriate number of classes. The class width of 10 is appropriate, and the number of classes (7) is reasonable (between 5 - 15, which is a common range for grouped frequency tables).

Answer:

The grouped frequency distribution table is as follows:

Class Interval	Frequency
20 - 29	7
30 - 39	5
40 - 49	2
50 - 59	1
60 - 69	2
70 - 79	1