Question

The graph illustrates the distribution of test scores taken by College Algebra students. The maximum possiblescore on the test was 110, while the mean score was 75 and the standard deviation was 7.54618968 75 82Distribution of Test Scores96QWhat is the approximate percentage of students who scored less than 61 on the test?What is the approximate percentage of students who scored between 61 and 89 on the test?What is the approximate percentage of students who scored between 68 and 75?What is the approximate percentage of students who scored between 75 and 96 on the test?

Answer 1

Assuming the distribution of the test scores is approximately normal, we calculate the percentages as cumulative probabilities of the bell curve.

To do that, we use Excel. You could use any other digital means or technology.

The z-score is defined as:

`z=(X-\mu)/(\sigma)`

Where X is the score whose probability we want to calculate, μ is the mean, and σ is the standard deviation.

1. Calculate the percentage of students who scored less than 61 on the test.

The z-score is:

`z=(61-75)/(7)=-2`

Now determine the probability p(z < -2) with Excel:

p(z < -2) = 0.02275

Multiplying by 100:

Required percentage = 2.275%

2. Calculate the percentage of students who scored between 61 and 89 on the test.

We calculate P(X < 61), P(X < 89) and subtract the results.

We have already calculated P(X < 61). Now calculate P(X < 89).

`z=(89-75)/(7)=2`

Now determine the probability p(z < 2) with Excel:

p(z < 2) = 0.97725

Multiplying by 100:

p(z < 2) = 97.725%

p(z < 2) - p(z < -2) = 97.725% - 2.275% = 95.450%

3) Calculate the percentage of students who scored between 68 and 75 on the test.

P(68 < X < 75) = P(X < 75) - P(X < 68)

Calculating the probabilities with Excel:

P(68 < X < 75) = 0.5 - 0.34134