The National Center for Educational Statistics reported that in 2009 the ACT college placement and admission
examination were normally distributed and had a mean of 21.1 and standard deviation of 5.1. A college
requires applicants to have an ACT score in the top 20% of all scores. Find the lowest ACT score a student
could get to meet this requirement. Of five students picked at random from those taking the ACT, find the
probability that none score high enough to satisfy the admission standard.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.20) and its Z score. Insert data into above equation.
1 - .20 = ?
To find the lowest ACT score a student could get to meet the top 20% requirement, we need to find the cutoff score corresponding to the 80th percentile.
To do this, we can use the standard normal distribution table or a calculator that provides the cumulative distribution function (CDF) for the normal distribution.
Using the standard normal distribution table:
1. Convert the ACT score to a z-score by using the formula: z = (x - mean) / standard deviation.
Substituting the given values, we have: z = (x - 21.1) / 5.1.
2. Look up the z-score in the standard normal distribution table to find the corresponding percentile. We need to find the z-score corresponding to the 80th percentile, which is the top 20%.
From the table, we find that the z-score corresponding to the 80th percentile is approximately 0.842.
3. Solve for x in the z-score formula: 0.842 = (x - 21.1) / 5.1.
Rearranging the formula, we have: x - 21.1 = 0.842 * 5.1.
Solve for x: x = 21.1 + 0.842 * 5.1.
Calculating the value, we find that the lowest ACT score to meet the requirement is approximately 25.31.
So, the lowest ACT score a student could get to meet the admission requirement is approximately 25.31.
To find the probability that none of five randomly selected students score high enough to satisfy the admission standard:
Assuming that ACT scores are independent and normally distributed among students:
1. Calculate the probability that a single randomly selected student does not score high enough to satisfy the admission standard.
This is equivalent to finding the probability that a randomly selected student has an ACT score below 25.31, i.e., finding the cumulative probability of a z-score less than 25.31.
Using the z-score formula from earlier, we have: z = (x - 21.1) / 5.1.
Substituting x = 25.31 and solving for z: (25.31 - 21.1) / 5.1 = 0.823.
Using the standard normal distribution table or a calculator, we find that the cumulative probability of a z-score less than 0.823 is approximately 0.7939.
Therefore, the probability that a single randomly selected student does not score high enough is approximately 0.7939.
2. Calculate the probability that none of the five randomly selected students score high enough.
Since each student's ACT score is independent of the others, the probability that none of them score high enough is the probability of one student not scoring high enough raised to the power of five (0.7939^5) since there are five students.
Calculating this, we find that the probability that none of the five randomly selected students score high enough is approximately 0.3290 or 32.9%.