Question

To be eligible for further consideration, applicants for certain civil service positions must first pass a written qualifying examination on which a score of 140 or more must be obtained. In a recent examination, it was found that the scores were normally distributed with a mean of 160 points and a standard deviation of 24 points. Determine the percentage of applicants who passed the written qualifying examination. (Round your answer to two decimal places.)

Answer 1

Solution :

Given ,

mean = = 160

standard deviation = = 24

P(x 140) = 1 - P(x< 140 )

= 1 - P[(x -) / < (140-160) /24 ]

= 1 - P(z <-0.5 )

Using z table

= 1 - 0.3085

= 0.6915

=69.15%

Answer 2

To determine the percentage of applicants who passed the written qualifying examination, we need to find the proportion of applicants who scored 140 or higher on the exam.

Given that the scores are normally distributed with a mean (μ) of 160 points and a standard deviation (σ) of 24 points, we can use the standard normal distribution (Z-distribution) to find the proportion.

Step 1: Calculate the Z-score for a score of 140 points.

The Z-score formula is:

Z = (X - μ) / σ

where X is the score, μ is the mean, and σ is the standard deviation.

Z = (140 - 160) / 24 Z = -20 / 24 Z ≈ -0.83

Step 2: Find the cumulative probability for a Z-score of -0.83 using a standard normal distribution table or a statistical calculator.

Using a standard normal distribution table or a calculator, we find that the cumulative probability for a Z-score of -0.83 is approximately 0.2033.

Step 3: Calculate the percentage of applicants who passed the examination.

The percentage of applicants who passed the examination is given by:

Percentage = (1 - Cumulative probability) * 100

Percentage = (1 - 0.2033) * 100 Percentage ≈ 79.67%

So, approximately 79.67% of applicants passed the written qualifying examination.