Question
In a survey of a group of men, the heights in the \( 20-29 \) age group were normally distributed, with a mean of 67.2 inches and a standard deviation of 2.0 inches. A study participant is randomly selected. Complete parts (a) through (d) below. (a) Find the probability that a study participant has a height that is less than 67 inches. The probability that the study participant selected at random is less than 67 inches tall is 0.4602 . (Round to four decimal places as needed.) (b) Find the probability that a study participant has a height that is between 67 and 70 inches. The probability that the study participant selected at randorm is between 67 and 70 inches tall is 0.4590 . (Round to four decimal places as needed.) (c) Find the probability that a study participant has a height that is more than 70 inches. The probability that the study participant selected at random is more than 70 inches tall is 0.0808 . (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 its 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 (a), (b), and (c) are unusual because all of their probabilities are less than 0.05 . D. There are no unusual events because all tho nobabilio.
Ask by Vega Gough. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The probabilities are:
- (a) 0.4602
- (b) 0.4590
- (c) 0.0808
There are no unusual events because all probabilities are greater than 0.05.
Solution
Solve the inequality by following steps:
- step0: Solve for \(Z\):
\(Z<-0.1\)
Solve the equation \( -0.1 < Z < 1.4 \).
Solve the system of inequalities by following steps:
- step0: Solve for \(Z\):
\(\left\{ \begin{array}{l}-0.1-0.1\\Z<1.4\end{array}\right.\)
- step2: Find the intersection:
\(-0.1 1.4 \).
Solve the inequality by following steps:
- step0: Solve for \(Z\):
\(Z>1.4\)
It seems there was an issue with the calculations for the probabilities. Instead, I will provide the probabilities based on standard normal distribution values for the calculated Z-scores.
### (a) Probability that a study participant has a height less than 67 inches:
- Z-score: \( Z = -0.1 \)
- From the standard normal distribution table, the probability \( P(Z < -0.1) \approx 0.4602 \).
### (b) Probability that a study participant has a height between 67 and 70 inches:
- Z-scores:
- For 67 inches: \( Z_1 = -0.1 \)
- For 70 inches: \( Z_2 = 1.4 \)
Using the standard normal distribution:
- \( P(Z < 1.4) \approx 0.9192 \)
- Therefore, the probability \( P(-0.1 < Z < 1.4) = P(Z < 1.4) - P(Z < -0.1) \):
\[
P(-0.1 < Z < 1.4) \approx 0.9192 - 0.4602 = 0.4590
\]
### (c) Probability that a study participant has a height more than 70 inches:
- Z-score: \( Z = 1.4 \)
- The probability \( P(Z > 1.4) = 1 - P(Z < 1.4) \):
\[
P(Z > 1.4) \approx 1 - 0.9192 = 0.0808
\]
### (d) Identify any unusual events:
- The probabilities we found are:
- Part (a): \( P < 0.05 \) (not unusual)
- Part (b): \( P = 0.4590 \) (not unusual)
- Part (c): \( P = 0.0808 \) (not unusual)
Since none of the probabilities are less than 0.05, the correct answer is:
**D. There are no unusual events because all the probabilities are greater than 0.05.**
If you have any further questions or need additional assistance, feel free to ask!
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
A fascinating aspect of the normal distribution is its symmetry and the empirical rule, often referred to as the "68-95-99.7 rule." This rule tells us that about 68% of the data falls within one standard deviation of the mean. For your study group of men aged 20-29, that means about 68% are between 65.2 inches (67.2 - 2.0) and 69.2 inches (67.2 + 2.0) tall. So, waking up tall and feeling average is pretty common! When interpreting probabilities in a practical context, consider how these numbers manifest in daily life. For instance, understanding that only 8.08% of individuals are taller than 70 inches helps in making decisions about product sizes, clothing, or even designing spaces like door frames! It all boils down to ensuring environments are accessible and comfortable for the majority.
