Question

A sample of final exam scores is normally distributed with a mean equal to 23 and a variance equal to 16.

Part (a)

What percentage of scores are between 19 and 27? (Round your answer to two decimal places.)

Part (b)

What raw score is the cutoff for the top 10% of scores? (Round your answer to one decimal place.)

Part (c)

What is the proportion below 17? (Round your answer to four decimal places.)

Part (d)

What is the probability of a score less than 29? (Round your answer to four decimal places.)

Answer 1

GIVEN:

Let X represents the final exam scores is normally distributed with a mean and a variance .

Thus standard deviation .

FORMULA USED:

To calculate the probability we will convert the raw score (X) into standard score (Z) using the formula given by,

SOLUTION:

PART (a): PERCENTAGE OF SCORES BETWEEN 19 AND 27:

The percentage of scores between 19 and 27 is,

  

  

  

{Since }

Using the z table, the first probability value is the value with corresponding row 1.0 and column 0.00 and the second probability value is the value with corresponding row -1.0 and column 0.00.

The percentage of scores between 19 and 27 is or %.

PART (b): CUTOFF SCORE FOR TOP 10% OF SCORE:

Given the probability of area to the right,

From the z table, for the probability 0.90, the corresponding row is 1.2 and column is 0.09. Thus the z score is .

Using the z score formula, the cutoff score for top 10% of scores is,

Thus the cutoff score for top 10% of scores is .

PART(c): PROPORTION OF SCORES BELOW 17:

The proportion of scores below 17 is,

  

  

Using the z table, the probability value is the value with corresponding row -1.5 and column 0.00.

  

The proportion of scores below 17 is .

PART(d): PROBABILITY OF SCORES LESS THAN 29:

The probability of scores less than 29 is,

  

  

Using the z table, the probability value is the value with corresponding row 1.5 and column 0.00.

  

The probability of scores less than 29 is .