Question

The time ( in minutes ) to fill out the teacher evaluation form for the course were as follow:

18, 19, 12, 18, 20, 15, 17, 19, 23, 30, 21, 16, 44, 17, 15, 12, 18, 26, 19, 12, 23, 22, 35, 16

(a) Find the range, the quartiles (Q1, Q3), the variance, the standard deviation.

(b) Draw a box-plot for the given data set.

(c) Find the percentile rank of the time of 20 minutes

Answer 1

a) Range = Maximum value - Minimum value

Range = 30 - 12

Range = 18

The sample size is n = 24.

We need to compute the 25% percentile based on the data provided.

Position	X (Asc. Order)
1	12
2	12
3	12
4	15
5	15
6	16
7	16
8	17
9	17
10	18
11	18
12	18
13	19
14	19
15	19
16	20
17	21
18	22
19	23
20	23
21	26
22	30
23	35
24	44

The next step is to compute the position (or rank) of the 25% percentile. The following is obtained:

Since the position found is not integer, the method of interpolation needs to be used. The 25% percentile is located between the values in the positions 6 and 7. Those values, based on the data organized in ascending order, are 16 and 16.

The value of 6.25 - 6 = 0.25 corresponds to the proportion of the distance between 16 and 16 where the percentile we are looking for is located at. In fact, we compute

This completes the calculation and we conclude that = 16.

The next step is to compute the position (or rank) of the 75% percentile. The following is obtained:

Since the position found is not integer, the method of interpolation needs to be used. The 75% percentile is located between the values in the positions 18 and 19. Those values, based on the data organized in ascending order, are 22 and 23.

The value of 18.75 - 18 = 0.75 corresponds to the proportion of the distance between 22 and 23 where the percentile we are looking for is located at. In fact, we compute

This completes the calculation and we conclude that = 22.75.

The sample size is n = 24. The provided sample data along with the data required to compute the variance are shown in the table below:

	X	X2
	18	324
	19	361
	12	144
	18	324
	20	400
	15	225
	17	289
	19	361
	23	529
	30	900
	21	441
	16	256
	44	1936
	17	289
	15	225
	12	144
	18	324
	26	676
	19	361
	12	144
	23	529
	22	484
	35	1225
	16	256
Sum =	487	11147

Also, the sample variance is

= 52.7064

Therefore, the sample standard deviation s is

s = 7.416

What percentile will be 20

20 lies at 16 observations

The next step is to compute the percentile of the 16th rank. Let it be xThe following is obtained:

Solving

25*x = 16*100

x = 64

64th percentile is the value 20