A suitable probability calculator can tell you the probability is about 6.7%.
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The standard deviation of the distribution of sample means is the population standard deviation (6) divided by the square root of the sample size (√144=12). Thus, your threshold is about (19.25-20)/(6/12) = -1.5 standard deviations from the mean.
Answer:
25%.
Step-by-step explanation:
When calculating percentile of a value within a sample population, you'll want to use what's called the z-score formula, notated as such:
z = (x – μ) / σ
Or, the z-score equals the difference of the x-value minus the mean (μ), divided by the standard deviation (σ). We can use the equation to sub in our numbers:
Since we are looking for scores above 80, what we want to find is the area under the normal distribution curve that is to the <em>right</em> of our value of 80, or the pink area illustrated in the attached picture.
For this, we can use the standard Z table to look up our z-score and find what percent of our chart we've filled, to the <em>left</em> of our value. You can find Z tables in algebra textbooks or online, like this one: http://www.z-table.com/
In this instance, our z-score gives us a resulting Z table percentile between .7454 (z-score of 0.66) and .7486 (z-score of 0.67). This means that, in our normal distribution with a mean of 70 and a standard deviation of 15, the area to the <em>left</em> of the value of 80 is about 74.6%.
So, now we know that our value of 80 makes up about 74.6% of our curve from the left. To find the area to the <em>right</em>, we just subtract from the total area, which is 1 in a normally distributed curve. This gives us 25.4%. Since we're looking for the percent of test takes scoring above our value, I'll round that down to 25%.
