Question

There are 46 students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of 5 min and a standard deviation of 4 min. (Round your answers to four decimal places.) A button hyperlink to the SALT program that reads: Use SALT. (a) If grading times are independent and the instructor begins grading at 6:50 P.M. and grades continuously, what is the (approximate) probability that he is through grading before the 11:00 P.M. TV news begins

Answer 1

0.7704 = 77.04% probability that he is through grading before the 11:00 P.M. TV news begins

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Distribution of n values from a normal distribution:

If we take n values from a normally distributed variable, the mean is
`\mu*n`, and the standard deviation is
`s = \sigma√(n)`

There are 46 students in an elementary statistics class. For each student, the mean is of 5 min and a standard deviation of 4 min.

This means that

`\mu = 46*5 = 230`

`s = 4√(46) = 27.13`

(a) If grading times are independent and the instructor begins grading at 6:50 P.M. and grades continuously, what is the (approximate) probability that he is through grading before the 11:00 P.M. TV news begins?

This is the probability that he finishes grading in 4 hours and 10 minutes, that is, 4*60 + 10 = 250 minutes, which is the pvalue of Z when X = 250.

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

In this distribution

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

`Z = (250 - 230)/(27.13)`

`Z = 0.74`

`Z = 0.74` has a pvalue of 0.7704

0.7704 = 77.04% probability that he is through grading before the 11:00 P.M. TV news begins