The time taken by a randomly selected applicant for a...

Name: Problem 88, Chapter 4: Probability with Applications in Engineering, Science, and Technology
Uploaded: 2021-04-25T23:07:21-08:00
Duration: 5 min 8 s
Channel: Kian Manafi
Description: The time taken by a randomly selected applicant for a mortgage to fill out a certain form has a normal distribution with mean value 10 min and standard deviation 2 min. If five individuals fill out a form on 1 day and six on another, what is the probability that the sample average amount of time taken on each day is at most 11 min?

The time taken by a randomly selected applicant for a mortgage to fill out a certain form has a normal distribution with mean value 10 min and standard deviation 2 min. If five individuals fill out a form on 1 day and six on another, what is the probability that the sample average amount of time taken on each day is at most 11 min?

The time taken by a randomly selected applicant for a mortgage to fill out a certain form has a
normal distribution with mean value 10 min and standard deviation 2 min. If five individuals fill
out a form on 1 day and six on another, what is the probability that the sample average amount of
time taken on each day is at most 11 min?

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve. It is defined by its mean and standard deviation, and many natural phenomena are often modeled as normal due to the central limit behavior of independent random variables.

Sampling Distribution of the Sample Mean

This refers to the distribution of sample means drawn from a population. When samples are taken from a normally distributed population, the sample mean itself is normally distributed regardless of the sample size, which allows for easier probabilistic calculations.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It is calculated as the population standard deviation divided by the square root of the sample size. This concept quantifies the variability of the sample mean from the population mean.

Z-Score Calculation

A z-score measures how many standard errors an observation (or a sample mean) is away from the mean. It is used to convert values to a standard normal distribution, making it possible to determine probabilities and percentiles associated with the observation.

Transcript

00:01 Okay, so we're given a normal distribution with the mean of 10 and a standard deviation of 2.

00:06 And what we're trying to find is the probability that the mean x value is at most 11 over two different days.

00:14 And so what i'm going to do is cut this up into day one and day two and find the probabilities on day 1 and day 2 and then go from there.

00:24 So on day one, we have five people filling out the form.

00:30 N is equal to 5.

00:31 And what i'm going to do is use something called a z score, which basically just transforms our normal distribution into the standard normal distribution, and transforms our maximum value or 11, into whatever that value would be on the standard distribution table.

00:50 So then we can just look at that table to find our probabilities.

00:53 So the z score is the mean x value minus our u value, divided by the standard deviation divided by the square root of the number of trials or of n.

01:08 And so if i do this for 11, we would have 11 minus 10 divided by our standard deviation, which is 2, divided by square root of 5, since we have 5 trials.

01:25 And this is equal to 1 over 2 divided by the square of 5 and now you can just simplify this to square to 5 over 2 and we can plug this into our probability so we say the probability of z being less than or equal to 1 sorry square to 5 over 2 this is equal to our probability of our x -main being less than equal to 11.

01:56 So now we can just use the standard normal distribution table to find our probability.

02:04 And so the last thing we're going to want to do is actually divide our square root of 5 by 2.

02:13 So we have a value to look at for the standard distribution table, standard normal distribution table.

02:21 So this is square to 5 over 2 is equal to 1 .1...

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