Varsity team try out. Assume the distribution of heights to be normal w/a mean of 64inches & standard deviation of 4 inches. only 15 students passed. how many students joined the try out?
To determine the number of students who joined the tryouts, we need to use the properties of a normal distribution.
Given that the distribution of heights is normal with a mean of 64 inches and a standard deviation of 4 inches, we can use the concept of z-scores to find the proportion of students who passed the tryouts.
A z-score measures how many standard deviations an individual data point is away from the mean. We can convert the height of each student into a z-score using the formula:
z = (x - μ) / σ
where:
- z is the z-score
- x is the height of the student
- μ is the mean height of the distribution (64 inches)
- σ is the standard deviation of the distribution (4 inches)
To find the proportion of students who passed the tryouts, we need to determine the z-score that corresponds to the height cutoff for passing. Let's assume that the height cutoff for passing is H inches.
To find the z-score corresponding to height H, we can use the formula:
H = μ + z * σ
Rearranging the formula to solve for z:
z = (H - μ) / σ
We have 15 students who passed, so the proportion of students who passed is 15 divided by the total number of students (P = 15 / N).
To find N, we will use the standard normal distribution table (or a calculator/tool that can calculate z-scores) to look up the z-score that corresponds to the proportion P.
Using the table or calculator, find the z-score that corresponds to the given proportion P. Let's assume the z-score is Z.
Finally, substitute the values into the formula and solve for N:
P = Φ(Z) = 15 / N
Using the z-table or calculator, look up the value of P (which corresponds to Z) to find the total number of students who joined the tryout (N).