Question

For an undisclosed year the american college test (act) had a approximately normal distribution with a mean score of 21 and a standard deviation of 5.34. calculate the lowest act score which is in the top 2% of the test takers that year. be sure to show how you arrived at your answer?

Answer 1

To calculate the lowest ACT score that is in the top 2% of test takers, find the z-score corresponding to the top 2%, convert it to a percentile, and then back to the ACT score.

Step-by-step explanation:

To calculate the lowest ACT score that is in the top 2% of the test takers, we need to find the z-score corresponding to the top 2% and then convert it back to the ACT score.

1. First, we find the z-score using the formula: z = (x - mean) / standard deviation.
2. Next, we convert the z-score to a percentile using a z-table or calculator. The top 2% corresponds to a z-score of approximately 2.0537.
3. Finally, we convert the z-score back to the ACT score using the formula: x = (z * standard deviation) + mean.

By plugging in the values of mean = 21, standard deviation = 5.34, and z = 2.0537, we can calculate the lowest ACT score which is in the top 2%.

Answer 2

To calculate the lowest ACT score in the top 2%, we need to find the z-score that corresponds to the 98th percentile and then use the z-score formula to calculate the ACT score.

Step-by-step explanation:

The z-score formula can be used to find the score that corresponds to a given percentage in a normal distribution. Given that we want to find the lowest ACT score in the top 2%, we need to find the z-score that corresponds to the 98th percentile. The formula for calculating the z-score is:

z = (X - μ) / σ

where X is the value, μ is the mean, and σ is the standard deviation.

First, we need to find the z-score that corresponds to the 98th percentile. Using a z-table or a calculator, we can find that the z-score for the 98th percentile is approximately 2.055. Now, we can rearrange the z-score formula to solve for X:

X = z * σ + μ

Substituting the values we know, X = 2.055 * 5.34 + 21, we can calculate the lowest ACT score in the top 2% to be approximately 31.11.