Question

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 5.55.5 and a standard deviation of 2.42.4. Answer parts ​(a)dash–​(d) below.

Answer 1

Answer:

a) P(x<4)=0.4140

b) P(4<x<6)=0.3289

c) P(x>8)=0.0640

d) B. There are no unusual events because all the probabilities are greater than 0.05.

Step-by-step explanation:

The question is incomplete.

Complete question:

"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 4.5 and a standard deviation of 2.3. 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 places as needed.) . (Round to four decimal

(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 decimal places as needed.) (Round to four

(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 places as needed.) (Round to four decimal

(d) ldentily any unusual events. Explain your reasoning Choose the correct answer below

A. The event in part (a) is unusual because its probability is less than 0.05.

B. There are no unusual events because all the probabilities are greater than 0.05.

C. The events in parts (a) and (c) are unusual because their probabilities are less than 0.05

D. The events in parts (a), (b), and (c) are unusual because all of their probabilities are less than 0.05."

a) The probability of x<4 is

$z=(4-4.5)/2.3=-0.2174\\\\P(x<4)=P(z<-0.2174)=0.4140$

b) The probability of 4<x<6 is

$z_1=-0.2174\\\\z_2=(6-4.5)/2.3=0.6522\\\\P(4<x<6)=P(-0.2174<z<0.6522)=P(z<0.6522)-P(z<0.2174)=0.7429-0.4140=0.3289$

c) The probability of x>8 is

$z=(8-4.5)/2.3=1.5217\\\\P(x>8)=P(z>1.5217)=0.0640$

d) B. There are no unusual events because all the probabilities are greater than 0.05.

Answer 2

Using the normal distribution concept, the probability of obtaining the outcome of the events mentioned are :

• P(X < 4) = 0.2660

• P(4 ≤ X ≤ 6) = 0.3165

• P(X > 8) = 0.1488

Given a normal distribution with the following parameters :

• Mean, μ = 5.5

• Standard deviation, σ = 2.4

Recall :

• Zscore = (X - μ) ÷ σ

1.) Probability of response fewer than 4 :

P(Z < Zscore) = P(Z < (4 - 5.5)/ 2.4) = P(Z < - 0.625)

Using a normal distribution table :

P(Z < - 0.625) = 0.2660

2.) Probability of response between 4 and 6 :

P(4 ≤ X ≤ 6) = P(Z ≤ (6 - 5.5)/2.4) - P(Z ≤ (4 - 5.5)/ 2.4)

P(4 ≤ X ≤ 6) = P(Z ≤ 0.2083) - P(Z ≤ - 0.625)

Using the normal distribution table :

P(4 ≤ X ≤ 6) = 0.5825 - 0.2660 = 0.3165

3.) Probability of response greater than 8 :

P(X > 8) = P(Z > (8 - 5.5)/2.4) = P(Z > 1.0417)

Using a normal distribution table :

P( > 1.0417) = 1 - P(Z < 1.0417) = 1 - 0.85122 = 0.1488

Therefore, the probability of the all three events are ; 0.2660, 0.3165 and 0.1488 respectively.

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