a simple random sample of 60 is drawn from a normally distributed population, and the mean is found to be 28, with a standard deviation of 5. which of the following values is within the 95% confidence interval (z-score = 1.96) for the population mean?

remember, the margin of error, me, can be determined using the formula $me = \frac{z \cdot s}{\sqrt{n}}$

- the value of 26, because its not greater than 26.7 and less than 29.3.
- the value of 27, because its greater than 26.7 and less than 29.3.
- the value of 32, because its greater than 23 and less than 33.
- the value of 34, because its not greater than 23 and less than 33.

Question

a simple random sample of 60 is drawn from a normally distributed population, and the mean is found to be 28, with a standard deviation of 5. which of the following values is within the 95% confidence interval (z-score = 1.96) for the population mean?

remember, the margin of error, me, can be determined using the formula $me = \frac{z \cdot s}{\sqrt{n}}$

- the value of 26, because its not greater than 26.7 and less than 29.3.
- the value of 27, because its greater than 26.7 and less than 29.3.
- the value of 32, because its greater than 23 and less than 33.
- the value of 34, because its not greater than 23 and less than 33.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Margin of Error Calculation", "Confidence Interval Construction" ], "new_concepts": [], "current_concepts": [ "Margin of Error Calculation", "Confidence Interval Construction" ] } ## Calculate the margin of error Using the **Margin of Error Calculation** knowledge point $$ \begin{aligned} &n = 60,\quad \bar{x} = 28,\quad s = 5,\quad z = 1.96\ &ME = \frac{z \cdot s}{\sqrt{n}} = \frac{1.96 \cdot 5}{\sqrt{60}} = \frac{9.8}{7.746} \approx 1.265 \end{aligned} $$ ## Construct the confidence interval Using the **Confidence Interval Construction** knowledge point $$ \begin{aligned} &\text{Lower Limit} = \bar{x} - ME = 28 - 1.265 \approx 26.735\ &\text{Upper Limit} = \bar{x} + ME = 28 + 1.265 \approx 29.265\ &\text{Interval} \approx (26.7, 29.3) \end{aligned} $$ ## Evaluate the given options Using the **Confidence Interval Construction** knowledge point $$ \begin{aligned} &26 otin (26.7, 29.3)\ &27 \in (26.7, 29.3)\ &32 otin (26.7, 29.3)\ &34 otin (26.7, 29.3) \end{aligned} $$

(A) The value of 26, because it's not greater than 26.7 and less than 29.3.

(B) The value of 27, because it's greater than 26.7 and less than 29.3.

(C) The value of 32, because it's greater than 23 and less than 33.

(D) The value of 34, because it's not greater than 23 and less than 33.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Confidence Interval Construction" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the margin of error

Construct the confidence interval

Evaluate the given options

Answer: