the ages of the winners of a cycling tournament are approximately bell-shaped. the mean age is 28.7 years, with a standard deviation of 3.4 years. the winner in one recent year was 34 years old.

(a) transform the age to a z-score.
(b) interpret the results.
(c) determine whether the age is unusual.

(a) transform the age to a z-score.
$z = 1.56$
(type an integer or decimal rounded to two decimal places as needed.)

(b) interpret the results.
an age of 34 is 1.56 standard deviation(s) above the mean.
(type an integer or decimal rounded to two decimal places as needed.)

(c) determine whether the age is unusual. choose the correct answer below.
a. no, this value is not unusual. a z-score between $-2$ and $2$ is not unusual.
b. yes, this value is unusual. a z-score outside of the range from $-2$ to $2$ is unusual.
c. yes, this value is unusual. a z-score between $-2$ and $2$ is unusual.
d. no, this value is not unusual. a z-score outside of the range from $-2$ to $2$ is not unusual.

Question

the ages of the winners of a cycling tournament are approximately bell-shaped. the mean age is 28.7 years, with a standard deviation of 3.4 years. the winner in one recent year was 34 years old.

(a) transform the age to a z-score.
(b) interpret the results.
(c) determine whether the age is unusual.

(a) transform the age to a z-score.
$z = 1.56$
(type an integer or decimal rounded to two decimal places as needed.)

(b) interpret the results.
an age of 34 is 1.56 standard deviation(s) above the mean.
(type an integer or decimal rounded to two decimal places as needed.)

(c) determine whether the age is unusual. choose the correct answer below.
a. no, this value is not unusual. a z-score between $-2$ and $2$ is not unusual.
b. yes, this value is unusual. a z-score outside of the range from $-2$ to $2$ is unusual.
c. yes, this value is unusual. a z-score between $-2$ and $2$ is unusual.
d. no, this value is not unusual. a z-score outside of the range from $-2$ to $2$ is not unusual.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Z-Score Calculation", "Z-Score Interpretation" ], "new_concepts": [], "current_concepts": [ "Z-Score Calculation", "Z-Score Interpretation", "Normal Distribution Outliers" ] } ## Calculate the z-score Using the *Z-Score Calculation* knowledge point $$ z = \frac{x - \mu}{\sigma} = \frac{34 - 28.7}{3.4} = \frac{5.3}{3.4} \approx 1.56 $$ ## Interpret the z-score Using the *Z-Score Interpretation* knowledge point An age of 34 is $1.56$ standard deviations above the mean. ## Determine if the value is unusual Using the *Z-Score Interpretation* knowledge point A value is considered unusual if its z-score is less than $-2$ or greater than $2$. Since $z = 1.56$ lies within the interval $[-2, 2]$, this value is not unusual. ## Question 1 $z = 1.56$ ## Question 2 An age of 34 is $1.56$ standard deviation(s) above the mean. ## Question 3 A. No, this value is not unusual. A z-score between -2 and 2 is not unusual.

B. Yes, this value is unusual. A z-score outside of the range from -2 to 2 is unusual.

C. Yes, this value is unusual. A z-score between -2 and 2 is unusual.

D. No, this value is not unusual. A z-score outside of the range from -2 to 2 is not unusual.

{ "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Statistics", "Z-Score Calculation" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the z-score

Interpret the z-score

Determine if the value is unusual

Answer:

Question 1

Question 2

Question 3