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 10were normally distributed, with a mean of 4.5 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 (b) Find the probability that a randomly selected study participant's response was between 4 and 6.(Round to four decimal places as needed.)(c) Find the probability that a randomly selected study participant's response was more than 8.(Round to four decimal places as needed.)(d) Identify any unusual events. Explain your reasoning. Choose the correct answer below.O A. There are no unusual events because all the probabilities are greater than 0.05.OB. The events in parts (a) (b), and (c) are unusual because all of their probabilities are less than 0.05.OC. The events in parts (a) and (c) are unusual because their probabilities are less than 0.05.OD. The event in part (a) is unusual because its probability is less than 0.05.Click to select your answer(s).

Answer 1

a) Probability that a randomly selected study participant's response was less than 4 = 0.4210

b) Probability that a randomly selected study participant's response was between 4 and 6 = 0.3050

c) Probability that a randomly selected study participant's response was more than 8 = 0.0810

d) Option A is correct.

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

Step-by-step explanation

Since this is a normal distribution, we will need to standardize the scores to find any probability using the normal distribution table or calculator.

The standardized score for any value is the value minus the mean then divided by the standard deviation.

z = (x - μ)/σ

z = standardized form of the score

x = the score

μ = mean of the distribution = 4.5

σ = standard deviation of the distribution = 2.5

a) Probability that a randomly selected study participant's response was less than 4

P (X < 4)

We will standardize 4

z = (x - μ)/σ

z = (4 - 4.5)/2.5 = (-0.5/2.5) = -0.20

P(X < 4) = P(z < -0.20)

Using the normal distribution table or the calculator

P(X < 4) = P(z < -0.20) = 0.4210

b) Probability that a randomly selected study participant's response was between 4 and 6.

P(4 < X < 6)

We will standardize 4 and 6

z = (x - μ)/σ

z = (4 - 4.5)/2.5 = (-0.5/2.5) = -0.20

z = (6 - 4.5)/2.5 = (1.5/2.5) = 0.60

P(4 < X < 6) = P(-0.20 < z < 0.60) = P(z < 0.60) - P(z < -0.20)

P(z < 0.60) = 0.7260

P(z < -0.20) = 0.4210

P(4 < X < 6)

= P(-0.20 < z < 0.60)

= P(z < 0.60) - P(z < -0.20)

= 0.7260 - 0.4210

= 0.3050

c) Probability that a randomly selected study participant's response was more than 8

P(x > 8)

We will standardize 8

z = (x - μ)/σ

z = (8 - 4.5)/2.5 = (3.5/2.5) = 1.40

P(x > 8) = P(z > 1.40) = 1 - P(z ≤ 1.40) = 1 - 0.919 = 0.0810

d) From the answers we have obtained, we can see that none of the answers are less than 0.05.

Hence,

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

Hope this Helps!!!