the scores of a standardized test are normally distribution with a mean of 312 and a standard deviation of 68. The bottom 40% of scores are designed 'Need improvement'. What is the cutoff score for those who 'Need improvement'?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the Z score related to the proportion (.40). Insert values into equation above to find the score.
To find the cutoff score for those who 'Need improvement,' we need to determine the value that corresponds to the bottom 40% of scores in a normal distribution.
Step 1: Find the z-score for the bottom 40% of scores
The z-score is a measure of how many standard deviations a particular value is from the mean. We can use the cumulative distribution function (CDF) of the normal distribution to find the z-score.
The CDF gives us the probability that a score is less than or equal to a given value. In this case, we want to find the z-score corresponding to the bottom 40% of scores.
Using a standard normal distribution table or a statistical calculator, you can find that the z-score corresponding to the bottom 40% is approximately -0.253.
Step 2: Calculate the cutoff score
Once we have the z-score, we can use it to find the corresponding score value in the distribution using the formula:
x = μ + (z * σ)
where:
x = cutoff score
μ = mean of the distribution
z = z-score
σ = standard deviation of the distribution
In this case, the mean (μ) is 312 and the standard deviation (σ) is 68. Plugging in the values, we have:
x = 312 + (-0.253 * 68)
Calculating this, we find:
x ≈ 312 - 17.204 = 294.796
Thus, the cutoff score for those who 'Need improvement' is approximately 294.796 when rounded to the appropriate decimal place.
Note: Depending on the rounding method used, the final cutoff score may vary slightly.