Step 1 :The null and alternative hypotheses for the hypothesis test are \(H_{0}: \mu_{d}=0 \text{ in. }\) and \(H_{1}: \mu_{d} \neq 0 \text{ in. }\) respectively.
Step 2 :The test statistic for a t-test is calculated as follows: \(t = \frac{\text{sample mean} - \text{population mean}}{\text{sample standard deviation} / \sqrt{\text{sample size}}}\). In this case, the population mean (under the null hypothesis) is 0.
Step 3 :The heights of mothers are [6.2066e+01, 5.6300e-01, 6.9000e-02, 6.3000e-02, 5.6700e-01, 6.4000e-02, 6.5000e-02, 6.2000e-02, 6.4000e-02, 5.0000e-01] and the heights of daughters are [6.5068e+01, 5.6300e-01, 6.9000e-02, 6.6000e-02, 6.7000e-02, 6.5000e-02, 5.6500e-01, 5.6800e-01, 6.6000e-02, 5.0000e-01].
Step 4 :The differences between the heights of mothers and daughters are [3.002e+00, 0.000e+00, 0.000e+00, 3.000e-03, -5.000e-01, 1.000e-03, 5.000e-01, 5.060e-01, 2.000e-03, 0.000e+00].
Step 5 :The sample mean of the differences is 0.3513999999999996, the sample standard deviation is 0.9737233465186879, and the sample size is 10.
Step 6 :Substituting these values into the formula for the t-test statistic, we get \(t = \frac{0.3513999999999996 - 0}{0.9737233465186879 / \sqrt{10}} = 1.1412115913170626\).
Step 7 :The test statistic for the hypothesis test is approximately \(\boxed{1.14}\).