Assume that the populations are normally distributed. (a)...

Assume that the populations are normally distributed. (a) Test whether $\mu_{1}<\mu_{2}$ at the $\alpha=0.05$ level of significance for the given sample data. (b) Construct a $95 \%$ confidence interval about $\mu_{1}-\mu_{2}$. $$ \begin{array}{lcc} & \text { Sample 1 } & \text { Sample 2 } \\ \hline n & 40 & 32 \\ \hline \bar{x} & 94.2 & 115.2 \\ \hline s & 15.9 & 23.0 \\ \hline \end{array} $$

Assume that the populations are normally distributed.
(a) Test whether $\mu_{1}<\mu_{2}$ at the $\alpha=0.05$ level of significance for the given sample data.
(b) Construct a $95 \%$ confidence interval about $\mu_{1}-\mu_{2}$.
$$
\begin{array}{lcc}
& \text { Sample 1 } & \text { Sample 2 } \\
\hline n & 40 & 32 \\
\hline \bar{x} & 94.2 & 115.2 \\
\hline s & 15.9 & 23.0 \\
\hline
\end{array}
$$

Key Concepts

Confidence Interval for Difference of Means

A confidence interval for the difference of two means provides a range of values within which the true difference between the population means is likely to lie, with a specified level of confidence (e.g., 95%). The interval is constructed using the sample means, standard deviations, and sizes, and it reflects the uncertainty inherent in the estimate from sample data.

Significance Level

The significance level, often denoted by alpha (?), is a threshold that determines when to reject the null hypothesis in hypothesis testing. A commonly used alpha level, such as 0.05, indicates a 5% risk of incorrectly rejecting the null hypothesis, thus balancing the need for evidence against the probability of a Type I error.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about the population based on sample data. It involves setting up a null hypothesis and an alternative hypothesis, then using a test statistic to determine whether the observed data are consistent with the null hypothesis. Decisions are made based on the significance level, which quantifies the risk of a Type I error.

Two-Sample t-Test

The two-sample t-test is used to compare the means of two independent groups, assuming that the data follow a normal distribution. This test evaluates whether the difference between the sample means is statistically significant, taking into account the variability within each group and the sample sizes. Its formulation often involves pooling variances if equal variances are assumed, or using an approximate method when variances are unequal.

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