The time needed to complete a final examination in a...

The time needed to complete a final examination in a particular college course is normally distributed with a mean of 80 minutes and a standard deviation of 10 minutes. Answer the following questions. a. What is the probability of completing the exam in one hour or less? b. What is the probability that a student will complete the exam in more than 60 minutes but less than 75 minutes? c. Assume that the class has 60 students and that the examination period is 90 minutes in length. How many students do you expect will be unable to complete the exam in the allotted time?

The time needed to complete a final examination in a particular college course is normally
distributed with a mean of 80 minutes and a standard deviation of 10 minutes. Answer the
following questions.
a. What is the probability of completing the exam in one hour or less?
b. What is the probability that a student will complete the exam in more than 60 minutes
but less than 75 minutes?
c. Assume that the class has 60 students and that the examination period is 90 minutes
in length. How many students do you expect will be unable to complete the exam in
the allotted time?

Key Concepts

Normal Distribution

A continuous probability distribution that is symmetric about its mean, with its shape determined by the mean (location) and standard deviation (spread). It is widely used to represent real-valued random variables whose distributions are not known but can be assumed to have a natural variability around an average value.

Standardization (Z-Score Transformation)

A technique used to convert any normal random variable into a standard normal variable with a mean of 0 and a standard deviation of 1. This transformation facilitates probability calculations by allowing the use of standard normal distribution tables or software to find probabilities.

Cumulative Distribution Function (CDF)

A function that describes the probability that a random variable takes on a value less than or equal to a specific value. In the context of the normal distribution, it allows for determining the probability of outcomes falling within a given interval.

Expected Value in Sampling

A fundamental concept in statistics that represents the long-run average outcome of a random variable. In sampling contexts, it is used to estimate the number of occurrences (or non-occurrences) in a sample by multiplying the probability of an event by the total number of trials or subjects.

Transcript

00:01 All right, so we're given for a certain college course at the mean time to complete.

00:05 The final examination is 80 minutes and it has a standard deviation of 10 minutes.

00:11 So let's just jot that down.

00:15 And we're assuming that this is a normal distribution.

00:19 So i'm just going to let x equal the normal random variable of the number of minutes.

00:29 That's not how you spell minutes.

00:31 There we go.

00:37 Okay.

00:40 All right.

00:41 So let's do some z source.

00:43 Questions.

00:45 We need to find the probability that the exam is completed in an hour or less, so probability that x is less than 60.

00:56 Let's z score this, which is going to be 60 minus 80, all over 10, which is a really easy calculation because they're all multiples of 10.

01:09 That's negative 2.

01:14 That gives us a probability that z is less than or equal to negative 2 .0.

01:24 Of looking at our table 0 .028.

01:33 So we're looking for the probability that z is less than equal negative 2 .0.

01:39 So our work here is done.

01:44 Alright, second, we need to find the probability that x is between, or rather that it takes between an hour and 75 minutes to complete the exam.

01:57 And once again, mean is 80, standard deviation is 10, so we can divide this into a lower z score zl, upper z score is u.

02:10 We already know zl is negative 2 .0 from the last problem.

02:15 This time it's going to be 75 minus 80, all over 10.

02:20 That's negative 5 over 10, which is negative 0 .50.

02:25 Or actually, yeah...

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