The ACT scores of applicants to college are approximately normally distributed with a mean of 20.3 and a standard deviation of 3.1. If a random sample of 20 applicants to college is chosen find the probability that their average ACT score will be 19 and 21.
68%!
Here is the method.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z scores.
To find the probability that the average ACT score of a random sample of 20 applicants will be between 19 and 21, we need to use the Central Limit Theorem and the properties of the normal distribution.
The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution, as long as the sample is random and large enough.
1. First, calculate the standard deviation of the sampling distribution, which is the standard deviation of the population divided by the square root of the sample size:
Standard deviation of the sampling distribution = standard deviation of the population / √sample size
= 3.1 / √20
≈ 0.693
2. Next, we need to convert the given average scores of 19 and 21 to z-scores. The z-score measures the number of standard deviations a data point is from the mean. We can use the z-score formula:
z = (x - μ) / σ
For 19:
z1 = (19 - 20.3) / 0.693
For 21:
z2 = (21 - 20.3) / 0.693
3. Now, we can find the probabilities corresponding to these z-scores using a standard normal distribution table or a calculator.
P(19) = ϕ(z1)
P(21) = ϕ(z2)
Note: ϕ(z) represents the cumulative distribution function (CDF) of the standard normal distribution.
4. Finally, calculate the probability between 19 and 21 by subtracting the probability of 19 from the probability of 21:
P(19 ≤ X ≤ 21) = P(21) - P(19)
Using these steps and a standard normal distribution table or calculator, you can find the probability that the average ACT score of a random sample of 20 applicants will be between 19 and 21.