Question

1) Consider a class with a normal distribution of grades of 5.5. The standard deviation is 0.5. Calculate the percentage of students who obtained a grade: A) between 5 and 6; B) above 6; C) below 5.

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A) between 5 and 6

1. Convert grades to z-scores: Z = \frac{X - \mu}{\sigma}

Z_1 = \frac{5 - 5.5}{0.5} = -1

Z_2 = \frac{6 - 5.5}{0.5} = 1

2. Calculate cumulative probabilities:

P(-1 \leq Z \leq 1) = P(Z \leq 1) - P(Z \leq -1)

0.8413 - 0.1587 = 0.6826

3. Convert to percentage:

68.26\%

B) above 6

1. Convert grade to z-score: Z = \frac{X - \mu}{\sigma}

Z = \frac{6 - 5.5}{0.5} = 1

2. Calculate cumulative probability:

P(Z > 1) = 1 - P(Z \leq 1)

1 - 0.8413 = 0.1587

3. Convert to percentage:

15.87\%

C) below 5

1. Convert grade to z-score: Z = \frac{X - \mu}{\sigma}

Z = \frac{5 - 5.5}{0.5} = -1

2. Use cumulative probability:

P(Z < -1) = 0.1587

3. Convert to percentage:

15.87\%

Answer:

A) 68.26\%

B) 15.87\%

C) 15.87\%

1. Convert grades to z-scores:

2. Calculate cumulative probabilities:

3. Convert to percentage:

B) above 6

1. Convert grade to z-score:

2. Calculate cumulative probability:

3. Convert to percentage:

C) below 5

1. Convert grade to z-score:

2. Use cumulative probability:

3. Convert to percentage:

Answer:

A) 68.26\%

B) 15.87\%

C) 15.87\%

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