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An art history professor assigns letter grades on a test according to the following scheme.A: Top 5% of scoresB: Scores below the top 5% and above the bottom 62%C: Scores below the top 38% and above the bottom 22%D: Scores below the top 78% and above the bottom 5%F: Bottom 5% of scoresScores on the test are normally distributed with a mean of 73.3 and a standard deviation of 9.7. Find the numerical limits for a B grade. Round your answers to the nearest whole number, if necessary.

Respuesta :

Answer:

Grade B score:

[tex]76 \leq x \leq 89[/tex]    

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 73.3

Standard Deviation, σ = 9.7

We are given that the distribution of score on test is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

B: Scores below the top 5% and above the bottom 62%

We have to find the value of x such that the probability is 0.62

[tex]P( X > x) = P( z > \displaystyle\frac{x - 73.3}{9.7})=0.62[/tex]  

[tex]= 1 -P( z \leq \displaystyle\frac{x - 73.3}{9.7})=0.62 [/tex]  

[tex]=P( z \leq \displaystyle\frac{x - 73.3}{9.7})=0.38 [/tex]  

Calculation the value from standard normal z table, we have,  

[tex]\displaystyle\frac{x - 73.3}{9.7} = 0.305\\\\x = 76.26[/tex]  

We have to find the value of x such that the probability is 0.05

[tex]P(X < 0.95) = \\\\P( X < x) = P( z < \displaystyle\frac{x - 73.3}{9.7})=0.95[/tex]

Calculation the value from standard normal z table, we have,  

[tex]\displaystyle\frac{x - 73.3}{9.7} = 1.645\\\\x = 89.26[/tex]  

Thus, the numerical value of score to achieve grade B is

[tex]76 \leq x \leq 89[/tex]

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