Final answer:
In a recent study on world happiness, we can calculate the probabilities of different response values. The probability of a response being less than 4 is 0.2644, the probability of a response being between 4 and 6 is 0.6141, and the probability of a response being more than 8 is 0.8810. None of these events are considered unusual.
Step-by-step explanation:
To answer part (a), we need to find the probability that a randomly selected study participant's response was less than 4. We can use the standard normal distribution to calculate this probability. First, we need to standardize the response value using the formula: Z = (x - mean) / standard deviation. Plugging in the values, we get Z = (4 - 5.4) / 2.2 = -0.636. Next, we can use a standard normal distribution table or a calculator to find the cumulative probability associated with a Z-score of -0.636. The probability is approximately 0.2644.
To answer part (b), we need to find the probability that a randomly selected study participant's response was between 4 and 6. Similarly, we can standardize the values to find the corresponding Z-scores. Z1 = (4 - 5.4) / 2.2 = -0.636 and Z2 = (6 - 5.4) / 2.2 = 0.273. We can then find the cumulative probabilities associated with these Z-scores and subtract them to find the desired probability. The probability is approximately 0.6141.
To answer part (c), we need to find the probability that a randomly selected study participant's response was more than 8. Again, we can standardize the value to find the Z-score. Z = (8 - 5.4) / 2.2 = 1.182. Using a standard normal distribution table or a calculator, we can find the cumulative probability associated with a Z-score of 1.182. The probability is approximately 0.8810.
Based on the probabilities calculated in parts (a), (b), and (c), we can identify any unusual events. An event is considered unusual if its probability is less than 0.05. Comparing the probabilities calculated, we can see that the events in parts (a), (b), and (c) all have probabilities greater than 0.05. Therefore, none of the events are considered unusual. The correct answer choice is B. There are no unusual events because all the probabilities are greater than 0.05.
The Question is incomplete. The complete question is
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 mean response was 5.4 with a standard deviation of 2.2.
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 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.