Question

The scores on the SAT verbal test in recent years follow approximately thenormal distribution distribution. Students get a mean score of 531 with astandard deviation of 103. Use technology to answer these questions.a. What is the proportion of students scoring under 400 (4 decimal positions)?b. What is the proportion of students scoring between 400 and 550? (4decimal positions)c. What is the proportion of students scoring over 550? (4 decimal positions)d. How high must a student score to place in the top 10% of all studentstaking the SAT? State answer as a whole number.

Answer 1

In this question, we are asked to find the proportion of students scoring under 400, between 400 and 550, and over 550 on the SAT verbal test. We are also asked to determine the score necessary to place in the top 10% of all test-takers. By using the mean score, standard deviation, and normal distribution table or calculator, we can find the answers to these questions.

Step-by-step explanation:

To answer these questions, we need to use the normal distribution table or a calculator with a normal distribution function. Let's solve each part:

a. Proportion of students scoring under 400: We want to find P(X < 400), where X is the SAT verbal test score. Using the mean (531) and standard deviation (103), we find the z-score for 400 and use the normal distribution table or a calculator to find the proportion. The z-score is (400 - 531) / 103 = -1.2748. Looking up the z-score in the table, we find the proportion to be approximately 0.1010.

b. Proportion of students scoring between 400 and 550: We want to find P(400 < X < 550). Again, we find the z-scores for 400 and 550 and use the normal distribution table or a calculator to find the proportion. The z-score for 400 is -1.2748, and the z-score for 550 is (550 - 531) / 103 = 0.1845. Using the table, we find the proportion to be approximately 0.5455.

c. Proportion of students scoring over 550: We want to find P(X > 550). Similar to part b, we find the z-score for 550 and use the table or calculator to find the proportion. The z-score is 0.1845, and the proportion is approximately 0.4281.

d. Score to place in the top 10%: We want to find the score that corresponds to the top 10% of all students. We can use the z-score to find this score. The z-score corresponding to the top 10% is approximately 1.282. Using this z-score and the formula z = (X - mean) / standard deviation, we can solve for X and find X = 1.282 * 103 + 531 = 668.566. Rounding up to the nearest whole number, a student must score at least 669 to place in the top 10%.

Answer 2

a) 0.1017

b) 0.8969

c) 0.4268

d) A Student to be taken in the top 10% must score at least 663 points

1) Considering the mean score of 531 and a Standard Deviation of 103 we can sketch out:

Note that to the right we are adding 1, 2 and 3 Standard Deviations. To the left, we are subtracting 1,2, and 3 Standard Deviations.

2) Examining the alternatives.

a) According to the bell Shape,

Making use of a Table of z score and finding the Probability we have:

`\begin{gathered} z=(x-\mu)/(\sigma) \\ z=(400-531)/(103) \\ z=-1.27\text{ } \end{gathered}`

The corresponding z score for that gives us 0.1017. So about 10.17% are under 400.

Similarly for these ones below:

b) What is the proportion of students scoring between 400 and 550? (4

decimal positions):

0.8969

c) What is the proportion of students scoring over 550?

0.4268

d) Considering the Top 10% is made up of students who scored 663

`\begin{gathered} z=(663-531)/(103) \\ z=1.28 \end{gathered}`

This z score corresponds to 10% of the scorers.

Hence, the answers are:

a) 0.1017

b) 0.8969

c) 0.4268

d) A Student to be taken in the top 10% must score at least 663 points