this simulation allows you to understand the habits of a larger group by looking at a smaller sample.

calculate the following based on the data provided:

* mean of the sample: $\bar{x} = $
* standard deviation of the sample: $s = \sqrt{\frac{\sum \left(f \cdot (x - \bar{x})^2\
ight)}{n-1}} = $
* margin of error for a 90% confidence level: $me_{90\%} = $

Question

this simulation allows you to understand the habits of a larger group by looking at a smaller sample.

calculate the following based on the data provided:

* mean of the sample: $\bar{x} = $
* standard deviation of the sample: $s = \sqrt{\frac{\sum \left(f \cdot (x - \bar{x})^2\
ight)}{n-1}} = $
* margin of error for a 90% confidence level: $me_{90\%} = $

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Sample Mean", "Sample Standard Deviation", "Margin of Error" ], "new_concepts": [], "current_concepts": [ "Sample Mean", "Sample Standard Deviation", "Margin of Error" ] } ## Calculate the class midpoints and sample size Using the *Sample Mean* knowledge point $$ \begin{aligned} &\text{Midpoints } (x): \ &1-5 \implies x_1 = 3 \ &6-10 \implies x_2 = 8 \ &11-15 \implies x_3 = 13 \ &16-20 \implies x_4 = 18 \ &21-25 \implies x_5 = 23 \ &\text{Sample size } (n): \ &n = \sum f = 30 + 25 + 18 + 12 + 15 = 100 \end{aligned} $$ ## Calculate the sample mean Using the *Sample Mean* knowledge point $$ \begin{aligned} &\sum (f \cdot x) = (30 \cdot 3) + (25 \cdot 8) + (18 \cdot 13) + (12 \cdot 18) + (15 \cdot 23) \ &\sum (f \cdot x) = 90 + 200 + 234 + 216 + 345 = 1085 \ &\bar{x} = \frac{\sum (f \cdot x)}{n} = \frac{1085}{100} = 10.85 \end{aligned} $$ ## Calculate the sample standard deviation Using the *Sample Standard Deviation* knowledge point $$ \begin{aligned} &\sum f(x - \bar{x})^2: \ &30 \cdot (3 - 10.85)^2 = 30 \cdot (-7.85)^2 = 1848.675 \ &25 \cdot (8 - 10.85)^2 = 25 \cdot (-2.85)^2 = 203.0625 \ &18 \cdot (13 - 10.85)^2 = 18 \cdot (2.15)^2 = 83.205 \ &12 \cdot (18 - 10.85)^2 = 12 \cdot (7.15)^2 = 613.47 \ &15 \cdot (23 - 10.85)^2 = 15 \cdot (12.15)^2 = 2214.3375 \ &\sum f(x - \bar{x})^2 = 4962.75 \ &s = \sqrt{\frac{4962.75}{100 - 1}} = \sqrt{50.128787} \approx 7.08 \end{aligned} $$ ## Calculate the margin of error Using the *Margin of Error* knowledge point $$ \begin{aligned} &z_{\alpha/2} \text{ for } 90\% = 1.645 \ &ME_{90\%} = z_{\alpha/2} \cdot \frac{s}{\sqrt{n}} = 1.645 \cdot \frac{7.08}{\sqrt{100}} = 1.645 \cdot 0.708 \approx 1.16 \end{aligned} $$ * **Mean of the sample:** $\bar{x} = 10.85$ * **Standard deviation of the sample:** $s = 7.08$ * **Margin of Error for a 90% confidence level:** $ME_{90\%} = 1.16$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Margin of Error" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the class midpoints and sample size

Calculate the sample mean

Calculate the sample standard deviation

Calculate the margin of error

Answer: