Questions: 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 6.3 and a standard deviation of 1.8 . Answer parts (a)-(d) below. (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 0.1725 . (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), (b), and (c) are unusual because all of their probabilities are less than 0.05. B. The events in parts (a) and (c) are unusual because their probabilities are less than 0.05 . C. The event in part (a) is unusual because its probability is less than 0 ov. 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 6.3 and a standard deviation of 1.8 . Answer parts (a)-(d) below. (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 0.1725 . (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), (b), and (c) are unusual because all of their probabilities are less than 0.05. B. The events in parts (a) and (c) are unusual because their probabilities are less than 0.05 . C. The event in part (a) is unusual because its probability is less than 0 ov. D. There are no unusual events because all the probabilities are greater than 0.05 .
Transcript 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 6.3 and a standard deviation of 1.8 . Answer parts (a)-(d) below. (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 0.1725 . (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), (b), and (c) are unusual because all of their probabilities are less than 0.05. B. The events in parts (a) and (c) are unusual because their probabilities are less than 0.05 . C. The event in part (a) is unusual because its probability is less than 0 ov. D. There are no unusual events because all the probabilities are greater than 0.05 .
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Solution

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Solution Steps

Step 1: Calculate the Z-Score

To find the Z-score for the response value of \(8\), we use the formula:

\[ z = \frac{X - \mu}{\sigma} \]

Substituting the values:

\[ z = \frac{8 - 6.3}{1.8} = \frac{1.7}{1.8} \approx 0.9444 \]

Thus, the Z-score for the response value \(8\) is \(0.9444\).

Step 2: Calculate the Probability

Next, we calculate the probability that a randomly selected study participant's response is more than \(8\). This is given by:

\[ P(X > 8) = P(Z > 0.9444) = \Phi(\infty) - \Phi(0.9444) \]

Using the cumulative distribution function \( \Phi \):

\[ P(X > 8) \approx 0.1725 \]

Step 3: Determine Unusual Events

To determine if the event of a response greater than \(8\) is unusual, we check the probability:

Since \(P(X > 8) = 0.1725\) is greater than \(0.05\), we conclude that this event is not unusual.

Final Answer

The probability that a randomly selected study participant's response was more than \(8\) is \(0.1725\), and the event is not unusual.

Thus, the answer is:

\(\boxed{B}\)

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