Question

There are 50 students in a calculus class. The amount of time needed for the instructor to grade a randomly chosen midterm exam paper is a random variable with a mean of 6 minutes and a standard deviation of 4 minutes. If grading times are independent, what is the probability that the instructor can finish grading in 4 and a half hours (round off to second decimal place)

Answer 1

14.46% probability that the instructor can finish grading in 4 and a half hours

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For sums of n values from a distribution, the mean is
`n\mu` and the standard deviation is
`s = √(n)\sigma`

In this problem, we have that:

`\mu = 6*50 = 300, s = √(50)*4 = 28.2843`

What is the probability that the instructor can finish grading in 4 and a half hours

Four and half hours is 4.5*60 = 270.

So this probability is the pvalue of Z when X = 270.

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (270 - 300)/(28.2843)`

`Z = -1.06`

`Z = -1.06` has a pvalue of 0.1446

14.46% probability that the instructor can finish grading in 4 and a half hours