The scores on an examination are normally distributed with a mean of 70 and a standard deviation of 6. Michael received a score on the examination between the lower quartile and the upper quartile of the scores. Which shaded region most accurately represents where Michael's score lies? A. B. C. D.

SSCE HSC Mathematics Standard Further statistical analysis: The scores on an examination are

The scores on an examination are normally distributed with a mean of 70 and a standard deviation of 6 - HSC - SSCE Mathematics Standard - Question 15 - 2019 - Paper 1

Question 15

The scores on an examination are normally distributed with a mean of 70 and a standard deviation of 6. Michael received a score on the examination between the lower ... show full transcript

Worked Solution & Example Answer:The scores on an examination are normally distributed with a mean of 70 and a standard deviation of 6 - HSC - SSCE Mathematics Standard - Question 15 - 2019 - Paper 1

Step 1

Determine the Lower Quartile (Q1)

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Answer

In a normal distribution, the lower quartile (Q1) corresponds to the 25th percentile. To find Q1, we can use the mean ( $ar{x} = 70$ ) and standard deviation ( $ext{SD} = 6$ ).

Using the z-score for the 25th percentile, approximately $z ext{ value} ext{ for } Q1 ext{ is } -0.67449.$

Using the formula: $Q1 = ar{x} + z imes ext{SD}$

Substituting the values: $Q1 = 70 + (-0.67449) imes 6 \\ = 70 - 4.04694 \\ ext{approximately } 65.95$

Step 2

Determine the Upper Quartile (Q3)

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Answer

In the same way, the upper quartile (Q3) corresponds to the 75th percentile. The z-score for the 75th percentile is approximately $z ext{ value} ext{ for } Q3 ext{ is } 0.67449.$

Using the formula: $Q3 = ar{x} + z imes ext{SD}$

Substituting the values: $Q3 = 70 + (0.67449) imes 6 \\ = 70 + 4.04694 \\ ext{approximately } 74.05$

Step 3

Identify the Shaded Region

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Answer

Michael's score lies between the lower quartile (approximately 65.95) and the upper quartile (approximately 74.05). Therefore, we need to identify the shaded region in the options that fall between these two values.

Thus, the correct answer is option A, which represents this range in the provided graph.

The scores on an examination are normally distributed with a mean of 70 and a standard deviation of 6 - HSC - SSCE Mathematics Standard - Question 15 - 2019 - Paper 1

Worked Solution & Example Answer:The scores on an examination are normally distributed with a mean of 70 and a standard deviation of 6 - HSC - SSCE Mathematics Standard - Question 15 - 2019 - Paper 1

Determine the Lower Quartile (Q1)

Determine the Upper Quartile (Q3)

Identify the Shaded Region

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