a clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. before treatment, 15 subjects had a mean wake time of 105.0 min. after treatment, the 15 subjects had a mean wake time of 82.2 min and a standard deviation of 21.7 min. assume that the 15 sample values appear to be from a normally distributed population and construct a 90% confidence interval estimate of the mean wake time for a population with drug treatments. what does the result suggest about the mean wake time of 105.0 min before the treatment? does the drug appear to be effective?

construct the 90% confidence interval estimate of the mean wake time for a population with the treatment.

$\square\text{ min}

Question

a clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. before treatment, 15 subjects had a mean wake time of 105.0 min. after treatment, the 15 subjects had a mean wake time of 82.2 min and a standard deviation of 21.7 min. assume that the 15 sample values appear to be from a normally distributed population and construct a 90% confidence interval estimate of the mean wake time for a population with drug treatments. what does the result suggest about the mean wake time of 105.0 min before the treatment? does the drug appear to be effective?

construct the 90% confidence interval estimate of the mean wake time for a population with the treatment.

\$\square\text{ min} < \mu < \square\text{ min}\$
(round to one decimal place as needed.)

construct the 90% confidence interval estimate of the mean wake time for a population with the treatment.

$\square\text{ min} < \mu < \square\text{ min}$
(round to one decimal place as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "One-Sample t Confidence Interval", "Critical Value t-score" ], "current_concepts": [ "One-Sample t Confidence Interval", "Critical Value t-score" ] } ## Identify the given sample statistics and parameters $$ \begin{aligned} n &= 15 \ \bar{x} &= 82.2 \ s &= 21.7 \ \text{Confidence Level} &= 0.90 \ \text{Degrees of Freedom } (df) &= n - 1 = 14 \end{aligned} $$ ## Determine the critical value $t_{\alpha/2}$ and margin of error $E$ $$ \begin{aligned} \alpha &= 1 - 0.90 = 0.10 \ t_{0.05, 14} &\approx 1.761 \ E &= t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} = 1.761 \cdot \frac{21.7}{\sqrt{15}} \approx 1.761 \cdot 5.6029 \approx 9.8667 \end{aligned} $$ ## Calculate the confidence interval limits $$ \begin{aligned} \text{Lower Limit} &= \bar{x} - E = 82.2 - 9.8667 = 72.3333 \approx 72.3 \ \text{Upper Limit} &= \bar{x} + E = 82.2 + 9.8667 = 92.0667 \approx 92.1 \end{aligned} $$ Construct the 90% confidence interval estimate of the mean wake time for a population with the treatment. 72.3 $\text{min} < \mu <$ 92.1 $\text{min}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "One-Sample t Confidence Interval" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given sample statistics and parameters

Determine the critical value \(t_{\alpha/2}\) and margin of error \(E\)

Calculate the confidence interval limits

Answer: