Question

Scores on a standardized reading test for fourth-gradestudents form a normal distribution with u= 60 ando= 20. What is the probability of obtaining a samplemean greater than M = 65 for each of the following?a. A sample of n=16 studentsb. A sample of n=25 studentse. A sample of n=100 students

Answer 1

Explanation

From the statement, we know that the population of scores of a test follow a normal distribution with:

• mean μ = 60,

,

• standard deviation σ = 20.

We want to compute the probability of obtaining a sample mean greater than M = 65 for different values of the size of the sample n.

First, the z-score for the sample mean with size n is given by:

$z(n)=\frac{M-\mu}{\sigma_S}=\sqrt{n}\cdot(\frac{M-\mu}{\sigma}).$

Where σₛ = σ/√n is the standard deviation of the sample.

The probability of obtaining a sample mean X greater than M, is given by:

$P(X>M)=P(Z>z).$

Where the probability P(Z > z) is obtained from a table for z-scores.

(a) Sample with n = 16 students

We compute the z-score with n = 16:

$z(16)=\sqrt{16}\cdot(\frac{65-60}{20})=1.$

Using a table for z-scores, we get:

$P(X>65)=P(Z>1)=0.15866.$

(b) Sample with n = 25 students

We compute the z-score with n = 25:

$z(25)=\sqrt{25}\cdot(\frac{65-60}{20})=1.25.$

Using a table for z-scores, we get:

$P(X>65)=P(Z>1.25)=0.10565.$

(c) Sample with n = 100 students

We compute the z-score with n = 100:

$z(100)=\sqrt{100}\cdot(\frac{65-60}{20})=2.5.$

Using a table for z-scores, we get:

$P(X>100)=P(Z>2.5)=0.0062097.$Answer

a. P(X > M = 65, n = 16) = 0.15866

b. P(X > M = 65, n = 25) = 0.10565

c. P(X > M = 65, n = 100) = 0.0062097