The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, normal distribution will appear as a bell curve.

Let's solve the problem:

(i) To find the number of workers earning more than Rs. 160, we first need to calculate the z-score. The z-score is a measure of how many standard deviations an element is from the mean.

Z = (X - μ) / σ

Where:
X = value = Rs. 160
μ = mean = Rs. 140
σ = standard deviation = Rs. 10

So, Z = (160 - 140) / 10 = 2

We look up this z-score in the z-table (or use a calculator with a normal distribution function) to find the area to the left under the curve, which is 0.9772. But we want the number of workers earning more than Rs. 160, so we need to find the area to the right under the curve, which is 1 - 0.9772 = 0.0228.

This means that 2.28% of workers earn more than Rs. 160. So, the number of workers is 0.0228 * 1000 = 22.8 ≈ 23 workers.

(ii) Similarly, to find the number of workers earning less than Rs. 126, we calculate the z-score:

Z = (126 - 140) / 10 = -1.4

The area to the left under the curve for Z = -1.4 is 0.0808, which means that 8.08% of workers earn less than Rs. 126. So, the number of workers is 0.0808 * 1000 = 80.8 ≈ 81 workers.

(iii) To find the number of workers earning between Rs. 140 and Rs. 144, we calculate the z-scores for both values:

Z1 = (140 - 140) / 10 = 0
Z2 = (144 - 140) / 10 = 0.4

The area to the left under the curve for Z = 0 is 0.5 (since 0 is the mean), and for Z = 0.4 is 0.6554. So, the area between these two z-scores is 0.6554 - 0.5 = 0.1554, which means that 15.54% of workers earn between Rs. 140 and Rs. 144. So, the number of workers is 0.1554 * 1000 = 155.4 ≈ 155 workers.

Question

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, normal distribution will appear as a bell curve.

Let's solve the problem:

(i) To find the number of workers earning more than Rs. 160, we first need to calculate the z-score. The z-score is a measure of how many standard deviations an element is from the mean.

Z = (X - μ) / σ

Where:
X = value = Rs. 160
μ = mean = Rs. 140
σ = standard deviation = Rs. 10

So, Z = (160 - 140) / 10 = 2

We look up this z-score in the z-table (or use a calculator with a normal distribution function) to find the area to the left under the curve, which is 0.9772. But we want the number of workers earning more than Rs. 160, so we need to find the area to the right under the curve, which is 1 - 0.9772 = 0.0228.

This means that 2.28% of workers earn more than Rs. 160. So, the number of workers is 0.0228 * 1000 = 22.8 ≈ 23 workers.

(ii) Similarly, to find the number of workers earning less than Rs. 126, we calculate the z-score:

Z = (126 - 140) / 10 = -1.4

The area to the left under the curve for Z = -1.4 is 0.0808, which means that 8.08% of workers earn less than Rs. 126. So, the number of workers is 0.0808 * 1000 = 80.8 ≈ 81 workers.

(iii) To find the number of workers earning between Rs. 140 and Rs. 144, we calculate the z-scores for both values:

Z1 = (140 - 140) / 10 = 0
Z2 = (144 - 140) / 10 = 0.4

The area to the left under the curve for Z = 0 is 0.5 (since 0 is the mean), and for Z = 0.4 is 0.6554. So, the area between these two z-scores is 0.6554 - 0.5 = 0.1554, which means that 15.54% of workers earn between Rs. 140 and Rs. 144. So, the number of workers is 0.1554 * 1000 = 155.4 ≈ 155 workers.

Knowee AI · Accepted Answer