A study was done using a treatment group and a placebo group. The results are shown in the table. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. Use a 0.05 significance level for both parts.
\begin{tabular}{|c|c|c|}
\hline & Treatment & Placebo \\
\hline$\mu$ & $\mu_{1}$ & $\mu_{2}$ \\
\hline $\mathrm{n}$ & 26 & 35 \\
\hline $\bar{x}$ & 2.33 & 2.66 \\
\hline s & 0.56 & 0.98 \\
\hline
\end{tabular}
populations with the same mean.
18. Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the two samples are from populations with the same mean.
c. Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that the two samples are from populations with the same mean.
(1) Reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that the two samples are from populations with the same mean.
b. Construct a confidence interval suitable for testing the claim that the two samples are from populations with the same mean.
$\square< \mu_{1}-\mu_{2}< \square$
(Round to three decimal places as needed.)

Step 1 ：Given the following data: sample sizes \(n_1 = 26\) and \(n_2 = 35\), sample means \(\bar{x}_1 = 2.33\) and \(\bar{x}_2 = 2.66\), and sample standard deviations \(s_1 = 0.56\) and \(s_2 = 0.98\). The significance level is \(\alpha = 0.05\).

Step 2 ：We are asked to perform a hypothesis test to determine if there is a significant difference between the means of the treatment and placebo groups. The null hypothesis is that the means of the two groups are equal, and the alternative hypothesis is that the means are not equal.

Step 3 ：We use a two-sample t-test since the samples are independent and we do not assume equal population standard deviations.

Step 4 ：We also need to construct a confidence interval for the difference between the means of the two groups. The formula for a confidence interval for the difference between two means is \((\bar{x}_1 - \bar{x}_2) \pm t \sqrt{(\frac{s_1^2}{n_1}) + (\frac{s_2^2}{n_2})}\), where \(s_1\) and \(s_2\) are the sample standard deviations, \(n_1\) and \(n_2\) are the sample sizes, and \(t\) is the t-score for the desired level of confidence.

Step 5 ：The t statistic is negative, which indicates that the mean of the treatment group is less than the mean of the placebo group. The p value is less than the significance level of 0.05, which means we reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the two samples are from populations with the same mean.

Step 6 ：The confidence interval for the difference between the means of the two groups is (-0.728, 0.068). Since this interval does not contain zero, we can conclude that there is a significant difference between the means of the two groups.

Step 7 ：Final Answer: \n1. Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the two samples are from populations with the same mean. \n2. The confidence interval for the difference between the means of the two groups is \(\boxed{(-0.728, 0.068)}\).