Question

scores on a standardized test are normally distributed with a mean of 75 and a standard deviation of 6. use technology or a standard normal table to find each proportion described.

1. what proportion of test-takers score more than 87?
2. what proportion of test-takers score less than 63?
3. what proportion score between 69 and 81 (between mean - 1 sd and mean + 1 sd)?

problem 4
scores on an iq test are normally distributed with a mean of 100 and a standard deviation of 15. use technology to find the iq scores that mark the boundaries for each group.

1. what iq score marks the 80th percentile (the score that 80% of test-takers score at or below)?
2. what iq score marks the 10th percentile (the score that 10% of test-takers score at or below)?
3. what two iq scores mark the boundaries of the middle 60% of test-takers? (hint: find the 20th and 80th percentiles.)

Explanation:

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First Set (Standardized Test, $\mu=75$, $\sigma=6$)

1. Score >87

Step1: Calculate z-score

$z = \frac{X-\mu}{\sigma} = \frac{87-75}{6} = 2$

Step2: Find upper tail proportion

Proportion = $1 - P(Z \leq 2) = 1 - 0.9772 = 0.0228$

2. Score <63

Step1: Calculate z-score

$z = \frac{X-\mu}{\sigma} = \frac{63-75}{6} = -2$

Step2: Find lower tail proportion

Proportion = $P(Z \leq -2) = 0.0228$

3. Score between 69 and 81

Step1: Calculate z-scores

$z_1 = \frac{69-75}{6} = -1$, $z_2 = \frac{81-75}{6} = 1$

Step2: Find middle proportion

Proportion = $P(Z \leq 1) - P(Z \leq -1) = 0.8413 - 0.1587 = 0.6826$

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Second Set (IQ Test, $\mu=100$, $\sigma=15$)

1. 80th percentile

Step1: Find z for 80th percentile

$z \approx 0.84$

Step2: Calculate IQ score

$X = \mu + z\sigma = 100 + (0.84)(15) = 112.6$

2. 10th percentile

Step1: Find z for 10th percentile

$z \approx -1.28$

Step2: Calculate IQ score

$X = \mu + z\sigma = 100 + (-1.28)(15) = 80.8$

3. Middle 60% boundaries

Step1: Find z for 20th/80th percentiles

$z_{20} \approx -0.84$, $z_{80} \approx 0.84$

Step2: Calculate boundary scores

$X_{20} = 100 + (-0.84)(15) = 87.4$, $X_{80} = 100 + (0.84)(15) = 112.6$

Answer:

1. Standardized Test Proportions:

• Score >87: 0.0228
• Score <63: 0.0228
• Score 69-81: 0.6826

1. IQ Test Boundaries:

• 80th percentile: 112.6
• 10th percentile: 80.8
• Middle 60% boundaries: 87.4 and 112.6