(a) find the 95% confidence interval of the mean for all drivers if the standard deviation of the population is 4.8. round your answers to one decimal place.

$20.5

Question

(a) find the 95% confidence interval of the mean for all drivers if the standard deviation of the population is 4.8. round your answers to one decimal place.

$20.5 < \mu < 23.5$

part 2 of 2

(b) find the 90% confidence interval of the mean for all drivers if the standard deviation of the population is 4.8. round your answers to one decimal place.

$\square < \mu < \square$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Confidence Interval", "Z-Confidence Interval for Mean" ], "new_concepts": [], "current_concepts": [ "Confidence Interval", "Z-Confidence Interval for Mean" ] } ## Extract sample statistics from Part (a) Using the **Confidence Interval** knowledge point $$ \begin{aligned} &\text{CI}_{95\%} = (20.5, 23.5)\ &\bar{x} = \frac{20.5 + 23.5}{2} = 22.0\ &E_{95\%} = \frac{23.5 - 20.5}{2} = 1.5 \end{aligned} $$ ## Determine sample size $n$ Using the **Z-Confidence Interval for Mean** knowledge point $$ \begin{aligned} &\sigma = 4.8,\quad z_{0.025} = 1.96\ &E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \implies 1.5 = 1.96 \cdot \frac{4.8}{\sqrt{n}}\ &\sqrt{n} = \frac{1.96 \cdot 4.8}{1.5} = 6.272\ &n = (6.272)^2 \approx 39.338 \end{aligned} $$ ## Calculate 90% confidence interval Using the **Z-Confidence Interval for Mean** knowledge point $$ \begin{aligned} &z_{0.05} = 1.645\ &E_{90\%} = 1.645 \cdot \frac{4.8}{\sqrt{39.338}} = 1.645 \cdot \frac{4.8}{6.272} \approx 1.259\ &\text{Lower Limit} = 22.0 - 1.259 = 20.741 \approx 20.7\ &\text{Upper Limit} = 22.0 + 1.259 = 23.259 \approx 23.3 \end{aligned} $$ (b) Find the 90% confidence interval of the mean for all drivers if the standard deviation of the population is 4.8. Round your answers to one decimal place. 20.7 $< \mu <$ 23.3 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Confidence Interval" ] }

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QUESTION IMAGE

Question

Explanation:

Extract sample statistics from Part (a)

Determine sample size \(n\)

Calculate 90% confidence interval

Answer: