A clinical trial was conducted to test the effectiveness of a drug for treating insomnia and other subjects. Before treatment, 13 subjects had a mean week time of 101.0 minutes. After treatment, the 13 subjects I mean wait time of 94.1 minutes and a standard deviation of 21.4 minutes. Assume that the 13 sample values appear to be from a normally distributed population and construct a 95% confidence interval estimate of the main wait time for a populations drug treatment what does a result suggest about the main week time of 101.0 minutes before the treatment? Does the drug appear to be effective?

Answer:

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Explanation:

To construct a 95% confidence interval estimate of the mean wait time for a population after the drug treatment, you can use the following formula for a confidence interval:

\[ \text{Confidence Interval} = \text{Sample Mean} \pm \left(\frac{\text{Standard Error}}{\sqrt{\text{Sample Size}}}\right) \times \text{Critical Value} \]

Here, you have the following information:

- Sample Mean (\( \bar{x} \)) after treatment = 94.1 minutes

- Standard Deviation (\( \sigma \)) after treatment = 21.4 minutes

- Sample Size (\( n \)) = 13

- Confidence Level = 95%

First, you need to find the critical value for a 95% confidence interval. This corresponds to a two-tailed confidence interval, so the critical value is based on the standard normal (Z) distribution. For a 95% confidence level, the critical Z-value is approximately ±1.96 (you can find this value from a Z-table or calculator).

Next, calculate the standard error (\(SE\)):

\[ SE = \frac{\sigma}{\sqrt{n}} \]

Substitute the values:

\[ SE = \frac{21.4}{\sqrt{13}} \approx 5.912 \text{ minutes} \]

Now, you can construct the confidence interval:

\[ \text{Confidence Interval} = 94.1 \pm (5.912 \times 1.96) \]

Calculating the endpoints:

Lower Limit = \( 94.1 - (5.912 \times 1.96) \)

Upper Limit = \( 94.1 + (5.912 \times 1.96) \)

Lower Limit ≈ 83.43 minutes

Upper Limit ≈ 104.77 minutes

The 95% confidence interval estimate for the mean wait time for the population after the drug treatment is approximately (83.43 minutes, 104.77 minutes).

Now, let's interpret the result:

- The original mean wait time before the treatment was 101.0 minutes.

- The lower limit of the confidence interval after the treatment is 83.43 minutes.

The result suggests that after the drug treatment, the mean wait time has decreased compared to before the treatment. The lower limit of the confidence interval is below the original mean wait time of 101.0 minutes. This suggests that the drug appears to be effective in reducing the mean wait time for the population.

However, it's essential to note that this is an observational study, and other factors could be at play. Further clinical trials and analysis are needed to establish the drug's effectiveness definitively.

Final answer:

The 95% confidence interval estimate for the mean wait time for a population's drug treatment is approximately (78.13, 109.07) minutes. The result suggests that the main wait time of 101.0 minutes before the treatment is not within the confidence interval, indicating that the drug appears to be effective in reducing the wait time.

Explanation:

To construct a 95% confidence interval estimate of the mean wait time for a population's drugtreatment, we can use the formula:

Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / √Sample Size)

Given that the sample mean after treatment is 94.1 minutes, the standard deviation is 21.4 minutes, and the sample size is 13, we can calculate the critical value using a t-distribution table or a statistical software.

Assuming a t-distribution with 12 degrees of freedom (n-1), the critical value for a 95% confidence level is approximately 2.179.

Substituting the values into the formula:

Confidence Interval = 94.1 ± (2.179) * (21.4 / √13)

Simplifying the expression:

Confidence Interval = 94.1 ± 15.97

Therefore, the 95% confidence interval estimate for the mean wait time for a population's drug treatment is approximately (78.13, 109.07) minutes.

The result suggests that the main wait time of 101.0 minutes before the treatment is not within the confidence interval. This indicates that the drug appears to be effective in reducing the wait time, as the confidence interval does not include the pre-treatment mean.

Learn more about confidence interval estimation and hypothesis testing here:

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