Answer :
Answer:
The minimum score an applicant must receive for admission is 392.
Step-by-step explanation:
Let the minimum score required be 'x₀'.
Given:
Mean score (μ) = 500
Standard deviation (σ) = 100
Percentage required for admission, P > 86% or 0.86
So, we are given the area under the normal distribution curve to the right of z-score which is 86%.
The z-score table gives the area left of the z-score value. So, we will find the z-score value for area 100 - 86 = 14% or 0.14
So, for value equal to 0.1401, the z-score = -1.08
Now, [tex]P(x>x_0)=P(z>-1.08)[/tex]
So, we find x₀ using the formula of z-score which is given as:
[tex]z=\frac{x_0-\mu}{\sigma}\\\\-1.08=\frac{x_0-500}{100}\\\\x_0-500=-1.08\times 100\\\\x_0=-108+500=392[/tex]
Therefore, the minimum score an applicant must receive for admission is 392.
Final answer:
To find the minimum score for admissions consideration, we need to find the score that corresponds to the 86th percentile of the normal distribution.
Explanation:
To find the minimum score an applicant must receive for admissions consideration, we need to find the score that corresponds to the 86th percentile of the normal distribution. First, we need to find the z-score corresponding to the 86th percentile using the z-table. From the z-table, the z-score corresponding to the 86th percentile is approximately 1.08. Next, we can use the z-score formula to find the minimum score:
x = Z * σ + μ
where x is the minimum score, Z is the z-score, σ is the standard deviation, and μ is the mean. Plugging in the values, we have:
x = 1.08 * 100 + 500 = 588
Therefore, the minimum score an applicant must receive for admissions consideration is 588.
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