Question

the table shows the distribution of the task completion time in a learning experiment by students in a class.

time (in minutes)	12.2	12.8	13.4	14.0	14.6	15.2	15.8	16.4
number of students	2	3	8	2	2	4	6	1

use this information to answer questions 23 and 24.

1. calculate the mean time.

a. 14.26
b. 14.37
c. 14.73
d. 14.53

1. if a student is chosen at random from the class, find the probability that the student spent more than the median time in completing the task.

a. $\frac{17}{28}$
b. $\frac{15}{28}$
c. $\frac{13}{28}$
d. $\frac{1}{2}$

1. if $p$ and $q$ are two simple statements, which of the following is equivalent to the statement: $prightarrow q$?

a. $sim qrightarrow p$
b. $sim prightarrow q$
c. $sim qrightarrow sim p$
d. $sim prightarrow sim q$

1. if the set $x = {0, 1, 2, 3, 4}$, find the number of proper subsets of $x$.

a. 32
b. 31
c. 25
d. 30

1. given that $(2x - 1)$ is one of the factors of $f(x)=4x^{3}-2x^{2}+px - 4$, find the value of $p$.

a. 6
b. 8
c. 12
d. 10

1. in how many ways can 2 blue, 3 red and 4 green balls be arranged in 9 holes?

a. 2,520
b. 1,260
c. 630
d. 960

Explanation:

Step1: Calculate the total number of students

Sum up the number of students in each time - interval. $2 + 3+8 + 2+2 + 4+6 + 1=28$.

Step2: Calculate the sum of the product of time and number of students

$(12.2\times2)+(12.8\times3)+(13.4\times8)+(14.0\times2)+(14.6\times2)+(15.2\times4)+(15.8\times6)+(16.4\times1)$
$=24.4 + 38.4+107.2+28+29.2+60.8+94.8+16.4$
$=399.2$.

Step3: Calculate the mean time

The mean $\bar{x}=\frac{\text{Sum of products}}{\text{Total number of students}}=\frac{399.2}{28}=14.26$.

Step4: Find the median

Since there are $n = 28$ (an even - numbered data set), the median is the average of the $14^{th}$ and $15^{th}$ ordered data values.
Counting the number of students in each interval: $2+3 + 8=13$ (first three intervals), and $2+3 + 8+2=15$ (first four intervals). So the median lies in the $14.0$ time - interval.

Step5: Calculate the number of students who spent more than the median time

The number of students who spent more than $14.0$ minutes is $2 + 4+6 + 1=13$.

Step6: Calculate the probability

The probability $P=\frac{\text{Number of students who spent more than the median}}{\text{Total number of students}}=\frac{13}{28}$.

Answer:

1. A. 14.26
2. C. $\frac{13}{28}$