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In a recent study on world happiness, participants were asked to evaluate their current lives on a scale from 0 to 10, where 0 represents the worst possible life and 10 represents the best possible life. The responses were normally distributed, with a mean of 6.1 and a standard deviation of 2.5. Answer parts (a)–(d) below. (a) Find the probability that a randomly selected study participant’s response was less than 4. The probability that a randomly selected study participant’s response was less than 4 is ______. (Round to four decimal places as needed.) (b) Find the probability that a randomly selected study participant’s response was between 4 and 6. The probability that a randomly selected study participant’s response was between 4 and 6 is ______. (Round to four decimal places as needed.) (c) Find the probability that a randomly selected study participant’s response was more than 8. The probability that a randomly selected study participant’s response was more than 8 is ______. (Round to four decimal places as needed.) (d) Identify any unusual events. Explain your reasoning. Choose the correct answer below. - A. The events in parts (a) and (c) are unusual because their probabilities are less than 0.05. - B. The event in part (a) is unusual because its probability is less than 0.05. - C. The events in parts (b), (a), and (c) are unusual because all of their probabilities are less than 0.05. - D. There are no unusual events because all the probabilities are greater than 0.05.

In a recent study on world happiness, participants were asked to evaluate their current lives on a scale from 0 to 10, where 0 represents the worst possible life and 10 represents the best possible life. The responses were normally distributed, with a mean of 6.1 and a standard deviation of 2.5. Answer parts (a)–(d) below. (a) Find the probability that a randomly selected study participant’s response was less than 4. The probability that a randomly selected study participant’s response was less than 4 is ______. (Round to four decimal places as needed.) (b) Find the probability that a randomly selected study participant’s response was between 4 and 6. The probability that a randomly selected study participant’s response was between 4 and 6 is ______. (Round to four decimal places as needed.) (c) Find the probability that a randomly selected study participant’s response was more than 8. The probability that a randomly selected study participant’s response was more than 8 is ______. (Round to four decimal places as needed.) (d) Identify any unusual events. Explain your reasoning. Choose the correct answer below. - A. The events in parts (a) and (c) are unusual because their probabilities are less than 0.05. - B. The event in part (a) is unusual because its probability is less than 0.05. - C. The events in parts (b), (a), and (c) are unusual because all of their probabilities are less than 0.05. - D. There are no unusual events because all the probabilities are greater than 0.05.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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In a recent study on world happiness, participants were asked to evaluate their current lives on a scale from 0 to 10, where 0 represents the worst possible life and 10 represents the best possible life. The responses were normally distributed, with a mean of 6.1 and a standard deviation of 2.5. Answer parts (a)–(d) below. (a) Find the probability that a randomly selected study participant’s response was less than 4. The probability that a randomly selected study participant’s response was less than 4 is ______. (Round to four decimal places as needed.) (b) Find the probability that a randomly selected study participant’s response was between 4 and 6. The probability that a randomly selected study participant’s response was between 4 and 6 is ______. (Round to four decimal places as needed.) (c) Find the probability that a randomly selected study participant’s response was more than 8. The probability that a randomly selected study participant’s response was more than 8 is ______. (Round to four decimal places as needed.) (d) Identify any unusual events. Explain your reasoning. Choose the correct answer below. - A. The events in parts (a) and (c) are unusual because their probabilities are less than 0.05. - B. The event in part (a) is unusual because its probability is less than 0.05. - C. The events in parts (b), (a), and (c) are unusual because all of their probabilities are less than 0.05. - D. There are no unusual events because all the probabilities are greater than 0.05.
Transcribed Image Text:In a recent study on world happiness, participants were asked to evaluate their current lives on a scale from 0 to 10, where 0 represents the worst possible life and 10 represents the best possible life. The responses were normally distributed, with a mean of 6.1 and a standard deviation of 2.5. Answer parts (a)–(d) below. (a) Find the probability that a randomly selected study participant’s response was less than 4. The probability that a randomly selected study participant’s response was less than 4 is ______. (Round to four decimal places as needed.) (b) Find the probability that a randomly selected study participant’s response was between 4 and 6. The probability that a randomly selected study participant’s response was between 4 and 6 is ______. (Round to four decimal places as needed.) (c) Find the probability that a randomly selected study participant’s response was more than 8. The probability that a randomly selected study participant’s response was more than 8 is ______. (Round to four decimal places as needed.) (d) Identify any unusual events. Explain your reasoning. Choose the correct answer below. - A. The events in parts (a) and (c) are unusual because their probabilities are less than 0.05. - B. The event in part (a) is unusual because its probability is less than 0.05. - C. The events in parts (b), (a), and (c) are unusual because all of their probabilities are less than 0.05. - D. There are no unusual events because all the probabilities are greater than 0.05.
Expert Solution
Step 1

It is given that the responses are normally distributed with mean 6.1 and standard deviation 2.5.

Denote the random variable X as the response of a randomly selected person.

That is, µ=6.1, σ = 2.5.

Thus, X~ N (6.1, 2.5).

 

a)

The probability that a randomly selected study participant's response was less than 4 is calculated as follows:

 

P(X<4) = P(Z<((4-6.1)/2.5) (by standardizing)

                  = P(Z<-0.84)            From the standard normal table, P(Z<-0.84) = 0.2005

                  = 0.2005

 

Thus, probability that a randomly selected study participant's response was less than 4 is 0.2005.

The probability value obtained using the Excel formula, =NORM.DIST(4,6.1,2.5,1) will be 0.2005.

Step 2

b)

The probability that a randomly selected study participant's response was between 4 and 6 is calculated as follows:

 

P(4<X<6) = P(Z<((6-6.1)/2.5) -P(Z<((4-6.1)/2.5) (by standardizing)

                     = P(Z<-0.04)- P(Z<-0.84)

From the standard normal table, P(Z<-0.04)= 0.4840 and P(Z<-0.84) = 0.2005

                     = 0.4840 – 0.2005

                     = 0.2835

 

Thus, the probability that a randomly selected study participant's response was between 4 and 6 is 0.2835.

The probability value obtained using the Excel formula, =NORM.DIST(6,6.1,2.5,1)- NORM.DIST(4,6.1,2.5,1) will be 0.2836.

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