in a survey of u.s. adults with a sample size of 2053, 339 said franklin roosevelt was the best president since world war ii. two u.s. adults are selected at random from this sample without replacement. complete parts (a) through (d).

(a) find the probability that both adults say franklin roosevelt was the best president since world war ii.
the probability that both adults say franklin roosevelt was the best president since world war ii is .
(round to three decimal places as needed.)

(b) find the probability that neither adult says franklin roosevelt was the best president since world war ii.
the probability that neither adult says franklin roosevelt was the best president since world war ii is .
(round to three decimal places as needed.)

(c) find the probability that at least one of the two adults says franklin roosevelt was the best president since world war ii.
the probability that at least one of the two adults says franklin roosevelt was the best president since world war ii is .
(round to three decimal places as needed.)

(d) which of the events can be considered unusual? explain. select all that apply.
the event in part (c) is unusual because its probability is less than or equal to 0.05.
the event in part (b) is unusual because its probability is less than or equal to 0.05.
the event in part (a) is unusual because its probability is less than or equal to 0.05.
none of these events are unusual.

Question

in a survey of u.s. adults with a sample size of 2053, 339 said franklin roosevelt was the best president since world war ii. two u.s. adults are selected at random from this sample without replacement. complete parts (a) through (d).

(a) find the probability that both adults say franklin roosevelt was the best president since world war ii.
the probability that both adults say franklin roosevelt was the best president since world war ii is  .
(round to three decimal places as needed.)

(b) find the probability that neither adult says franklin roosevelt was the best president since world war ii.
the probability that neither adult says franklin roosevelt was the best president since world war ii is  .
(round to three decimal places as needed.)

(c) find the probability that at least one of the two adults says franklin roosevelt was the best president since world war ii.
the probability that at least one of the two adults says franklin roosevelt was the best president since world war ii is  .
(round to three decimal places as needed.)

(d) which of the events can be considered unusual? explain. select all that apply.
  the event in part (c) is unusual because its probability is less than or equal to 0.05.
  the event in part (b) is unusual because its probability is less than or equal to 0.05.
  the event in part (a) is unusual because its probability is less than or equal to 0.05.
  none of these events are unusual.

Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Sampling Without Replacement", "Dependent Events", "At Least One Probability", "Unusual Event Probability" ], "new_concepts": [], "current_concepts": [ "Sampling Without Replacement", "Dependent Events", "At Least One Probability", "Unusual Event Probability" ] } ## Identify the given parameters Using the *Sampling Without Replacement* knowledge point $$ \begin{aligned} &N = 2053 \quad \text{(Total sample size)} \ &S = 339 \quad \text{(Number who said FDR was the best)} \ &F = N - S = 1714 \quad \text{(Number who did not say FDR was the best)} \ &n = 2 \quad \text{(Number of adults selected)} \end{aligned} $$ ## Calculate the probability for part (a) Using the *Dependent Events* knowledge point $$ \begin{aligned} P(\text{Both say FDR}) &= \frac{S}{N} \times \frac{S - 1}{N - 1} \ &= \frac{339}{2053} \times \frac{338}{2052} \ &= \frac{114582}{4212756} \approx 0.027199 \approx 0.027 \end{aligned} $$ ## Calculate the probability for part (b) Using the *Dependent Events* knowledge point $$ \begin{aligned} P(\text{Neither says FDR}) &= \frac{F}{N} \times \frac{F - 1}{N - 1} \ &= \frac{1714}{2053} \times \frac{1713}{2052} \ &= \frac{2936082}{4212756} \approx 0.69695 \approx 0.697 \end{aligned} $$ ## Calculate the probability for part (c) Using the *At Least One Probability* knowledge point $$ \begin{aligned} P(\text{At least one says FDR}) &= 1 - P(\text{Neither says FDR}) \ &= 1 - 0.69695 \approx 0.30305 \approx 0.303 \end{aligned} $$ ## Determine unusual events for part (d) Using the *Unusual Event Probability* knowledge point $$ \begin{aligned} &\text{An event is unusual if its probability is } \le 0.05. \ &P(\text{Part a}) \approx 0.027 \le 0.05 \quad \text{(Unusual)} \ &P(\text{Part b}) \approx 0.697 > 0.05 \quad \text{(Not unusual)} \ &P(\text{Part c}) \approx 0.303 > 0.05 \quad \text{(Not unusual)} \end{aligned} $$ ## Question 1 The probability that both adults say Franklin Roosevelt was the best president since World War II is 0.027. ## Question 2 The probability that neither adult says Franklin Roosevelt was the best president since World War II is 0.697. ## Question 3 The probability that at least one of the two adults says Franklin Roosevelt was the best president since World War II is 0.303. ## Question 4 The event in part (c) is unusual because its probability is less than or equal to 0.05.

The event in part (b) is unusual because its probability is less than or equal to 0.05.

The event in part (a) is unusual because its probability is less than or equal to 0.05.

None of these events are unusual.

{ "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Statistics", "Dependent Events" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given parameters

Calculate the probability for part (a)

Calculate the probability for part (b)

Calculate the probability for part (c)

Determine unusual events for part (d)

Answer:

Question 1

Question 2

Question 3

Question 4