Question

1. What is the z-score associated with the 75th percentile?

2. What z-scores bound the middle 50% of a normal distribution?

3. What z-score has 10% of the distribution above it?

4. What z-score has 20% of the distribution below it?

5. Reading comprehension scores for junior high students are normally distributed with a mean of80.0 and a standard deviation of 5.0.
a. What percent of students have scores greater than 87.5?

b. What percent of students have scores between 75 and 85

Answer 1

Solution :

Given that,  

Using standard normal table ,

1.

P(Z < z) = 75%

P(Z < 0.674) = 0.75

z = 0.674

z-score = 0.674

2.

Middle 50% the z value are

= 1 - 0.5 = 0.5

/ 2 = 0.25

1 - / 2 = 1 - 0.25 = 0.75

Z = 0.674

z-score = 0.674

3.

P(Z > z) = 10%

1 - P(Z < z) = 0.1

P(Z < z) = 1 - 0.1

P(Z < 1.282) = 0.9

z = 1.282

z-score = 1.282

4.

P(Z < z) = 20%

P(Z < -0.842) = 0.2

z = -0.842

z-score = -0.842

5.

a.

mean = = 80.0

standard deviation = = 5.0

P(x > 87.5) = 1 - P(x < 87.5)

= 1 - P[(x - ) / < (87.5 - 80.0) / 5.0]

= 1 - P(z < 1.5)

= 1 - 0.9332

= 0.0668

percent = 6.68%

b.  

P(75 < x < 85) = P[(75 - 80.0)/ 5.0) < (x - ) /  < (85 - 80.0) / 5.0) ]

= P(-1 < z < 1)

= P(z < 1) - P(z < -1)

= 0.8413 - 0.1587

= 0.6826

percent = 68.26%