Two independent random samples are taken from two populations. The results of these samples are summarized in the next table.

1. Form a 90% confidence interval for\({\mu _1} - {\mu _2}\)

The sample sizes are 135 and 148

The means are 12.2 and 8.3.

The variances are 2.1 and 3.0.

A confidence interval is an interval centered about the sample statistic with width equal to twice the margin of error. If many sample are taken from a population with the sample size then the proportion of the constructed confidence intervals that will contain the population parameter is\(1 - \alpha \) .

The confidence interval computed as

\(\begin{aligned}{l}CI &= \left( {{{\bar x}_1} - {{\bar x}_2}} \right) \pm {z_{\frac{\alpha }{2}}}\left( {\sqrt {\frac{{s_1^2}}{{{n_1}}} + \frac{{s_2^2}}{{{n_2}}}} } \right)\\ &= \left( {12.2 - 8.3} \right) \pm {z_{0.05}}\left( {\sqrt {\frac{{2.1}}{{135}} + \frac{3}{{148}}} } \right)\\ &= 3.9 \pm 1.645\left( {.189} \right)\\ = 3.9 \pm .310\\ &= \left( {3.9 - .310,3.9 + .310} \right)\\ &= \left( {3.59,4.21} \right)\end{aligned}\)

Therefore, the confidence interval is (3.59, 4.21)