Lesson 5.2: Inference for Difference in Means from Two Independent Groups

Software Lab 5.2 Solutions

Calculated value: $t=\dfrac{0.401}{0.268} = 1.496$ . Test output value: $1.50$ .

Figure 1: Independent samples t-test for ncbirths
Calculated value: $df = \dfrac{\left( \dfrac{1.43^2}{100}+\dfrac{1.60^2}{50} \right)^2}{\dfrac{1}{99}\left( \dfrac{1.43^2}{100} \right)^2 + \dfrac{1}{49}\left( \dfrac{1.60^2}{50} \right)^2} = 88.9$ . Test output value: $89.3$ .

Figure 2: Group descriptives for North Carolina births data
Calculated p-value: $0.1382$ . Test output value: $0.138$ .
The p-value is larger than the significance value, 0.05, so we do not reject the null hypothesis. There is insufficient evidence to say there is a difference in average birth weight of newborns from North Carolina mothers who did smoke during pregnancy and newborns from North Carolina mothers who did not smoke during pregnancy.
Calculated value: $t=\dfrac{-0.459}{0.101} = -4.545$ . Test output value: $-4.52$ .

Figure 3: Independent samples t-test for BMI and activity
Calculated value: $df = \dfrac{\left( \dfrac{4.74^2}{8342}+\dfrac{5.52^2}{4022} \right)^2}{\dfrac{1}{8341}\left( \dfrac{4.74^2}{8342} \right)^2 + \dfrac{1}{4021}\left( \dfrac{5.52^2}{4022} \right)^2} = 6964$ . Test output value: $6959$ .

Figure 4: Group descriptives for high-schooler physical activity
Calculated p-value: $0.000006$ . Test output value: $< 0.001$ .
The p-value is smaller than the significance value, 0.05, so we reject the null hypothesis in favour of the alternative hypothesis. There is sufficient evidence to say there is a difference in average body mass index of high schoolers who are physically active at least three days a week and high schoolers who are physically active two or fewer days a week.
$(\overline{y}_1-\overline{y}_2) \pm t^* \times \sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}} = -0.459 \pm 1.96(0.101) = -0.459 \pm 0.198$ $= (-0.657, -0.261)$ . The interval in the “Independent Samples T-Test” output is given as $(-0.657, -0.260)$ .

Figure 5: Independent samples confidence interval for high-schooler physical activity
We’re 95% confident the difference in average body mass index of high-schoolers who are physically active at least three days a week and high-schoolers who are physically active two or fewer days a week is between $-0.657$ and $-0.260$ . Although statistically significant, this difference is probably not of much practical significance.

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