Question

A test was given to 60 individuals. Half answered 70% of the questions correctly, and the other half answered 90% correctly. Which of the following statements is correct? Group of answer choices Mean < Median Mean = Median Standard Deviation = 0 Mean > Median

Answer 1

Answer:

Mean = Median.

Step-by-step explanation:

The number of students in the class is, n = 60.

It is provided that half of the students answered 70% of the questions correctly, and the other half answered 90% correctly.

Let X = number of correctly answered questions.

Compute the probability of correctly answering a question as follows:

$P=\frac{n_{1}p_{1}+n_{2}p_{2}}{n}=\frac{(30\times0.70)+(30\times0.90)}{60}=0.80$

The random variable X follows a Binomial distribution with parameters n = 60 and p = 0.80.

The mean of the Binomial distribution is:

$E(X)=np=60\times0.80=48$

So the mean of the random variable X is 48.

The median value of a data is the below which 50% of the distribution lies.

Let x denote the median value of the distribution of X.

As,

• np = 48 > 10
• n (1 - p) = 60 × (1 - 0.80) = 12 > 10

A normal distribution can be used to approximate the Binomial distribution.

Compute the value of x such that P (X < x) = 0.50 as follows:

$P(X<x)=0.50\\P(\frac{X-\mu}{\sigma}<\frac{x-np}{\sqrt{npq}})=0.50\\P(Z<z)=0.50$

The value of z for which P (Z < z) = 0.50 is 0.

The value of x is:

$z=\frac{x-np}{\sqrt{npq}}\\0=\frac{x-48}{\sqrt{9.6}}\\0=x-48\\x=48$

The median value of X is 48.

Thus, Mean = Median.