APFIVE The table provides summary statistics for two independent samples. To construct a 95% confidence interval for the difference between the two population means, which of the following is a necessary step, given that the population standard deviations are unknown?
| Group | Sample Size | Sample Mean | Sample Standard Deviation |
|---|---|---|---|
| Department 1 | 30 | 12.3 | 3.5 |
| Department 2 | 30 | 7.6 | 3.4 |
Use the t-distribution with degrees of freedom approximated by taking the minimum of $$n_1-1$$ and $$n_2-1$$ and compute the standard error as $$\sqrt{(s_1^2/n_1) + (s_2^2/n_2)}$$.
Calculate a pooled variance and then use the z-distribution with degrees of freedom equal to $$n_1+n_2-1$$.
Use a chi-square distribution to adjust for the variance in the sample standard deviations.
Use the normal distribution regardless of sample sizes when the population standard deviations are unknown.
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