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A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjocts. Before treatment, 20 subjects had a mean wake time of 101.0 min. After treatment, the 20 suby mean wake tme of 78.7min and a standard deviation of 23.2min. Assume that the 20 sample values appear to be from a normally distributed population and construct a 90% confidence interval es mean wake time for a population with drug treatrents. What does the result suggest about the mean wake time of 101.0 min before the treatment? Does the drug appear to be eflective? Construct the 90% confidence interval estimate of the mean wake time for a population with the treatment. ◻min < mu < ◻min (Round to one docimal place as needed.)

A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjocts. Before treatment, 2020 subjects had a mean wake time of 101101.00 min. After treatment, the 2020 suby mean wake tme of 78.7 min 78.7 \mathrm{~min} and a standard deviation of 23.2 min 23.2 \mathrm{~min} . Assume that the 2020 sample values appear to be from a normally distributed population and construct a 90% 90 \% confidence interval es mean wake time for a population with drug treatrents. What does the result suggest about the mean wake time of 101101.00 min before the treatment? Does the drug appear to be eflective?\newlineConstruct the 90% 90 \% confidence interval estimate of the mean wake time for a population with the treatment.\newline \square \min <\mu<\square \min \newline(Round to one docimal place as needed.)

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Q. A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjocts. Before treatment, 2020 subjects had a mean wake time of 101101.00 min. After treatment, the 2020 suby mean wake tme of 78.7 min 78.7 \mathrm{~min} and a standard deviation of 23.2 min 23.2 \mathrm{~min} . Assume that the 2020 sample values appear to be from a normally distributed population and construct a 90% 90 \% confidence interval es mean wake time for a population with drug treatrents. What does the result suggest about the mean wake time of 101101.00 min before the treatment? Does the drug appear to be eflective?\newlineConstruct the 90% 90 \% confidence interval estimate of the mean wake time for a population with the treatment.\newline□min⁡<μ<□min⁡ \square \min <\mu<\square \min \newline(Round to one docimal place as needed.)
  1. Identify Parameters: Identify the sample mean (xˉ\bar{x}), the sample standard deviation (ss), and the sample size (nn).\newlineSample mean (xˉ\bar{x}) after treatment = 78.778.7 min\newlineSample standard deviation (ss) = 23.223.2 min\newlineSample size (nn) = 2020
  2. Determine Confidence Level: Determine the appropriate z-score for the 9090% confidence level. Since the sample size is small n < 30 and the population standard deviation is unknown, we should use the t-distribution. However, for a 9090% confidence interval and a sample size of 2020, the t-score is very close to the z-score. For simplicity, we will use the z-score for a 9090% confidence interval, which is approximately 1.6451.645. Note that for more accurate results, especially with small sample sizes, the t-score should be used.\newlineZ-score for 9090% confidence = 1.6451.645
  3. Calculate SEM: Calculate the standard error of the mean (SEM), which is the standard deviation of the sampling distribution of the sample mean.\newlineSEM=snSEM = \frac{s}{\sqrt{n}}\newlineSEM=23.220SEM = \frac{23.2}{\sqrt{20}}\newlineSEM≈23.24.472SEM \approx \frac{23.2}{4.472}\newlineSEM≈5.188SEM \approx 5.188
  4. Calculate Margin of Error: Calculate the margin of error (ME) using the z-score and the standard error of the mean. \newlineME=z×SEMME = z \times SEM\newlineME=1.645×5.188ME = 1.645 \times 5.188\newlineME≈8.534ME \approx 8.534
  5. Construct Confidence Interval: Construct the confidence interval by subtracting and adding the margin of error from the sample mean.\newlineLower limit = xˉ−ME\bar{x} - ME\newlineLower limit = 78.7−8.53478.7 - 8.534\newlineLower limit ≈70.2\approx 70.2 (rounded to one decimal place)\newlineUpper limit = xˉ+ME\bar{x} + ME\newlineUpper limit = 78.7+8.53478.7 + 8.534\newlineUpper limit ≈87.2\approx 87.2 (rounded to one decimal place)
  6. Interpret Results: Interpret the confidence interval and compare it to the mean wake time before the treatment. The 90%90\% confidence interval for the mean wake time after treatment with the drug is between 70.2 min70.2\,\text{min} and 87.2 min87.2\,\text{min}. Since the interval does not include the mean wake time before treatment (101.0 min101.0\,\text{min}), this suggests that the drug treatment has effectively reduced the mean wake time, indicating that the drug appears to be effective for treating insomnia.

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