In 1990 , the mean height of women 20 years of age or older was 63.7 inches based on data obtained from the CDC. Suppose that a random sample of 45 women who are 20 years of age or older today results in a mean height of 63.9 inches and a sample standard deviation of 1.42 inches. Assume that the assumptions are met to see if women have gotten taller. \[ \begin{array}{l} H_{0}: \mu=63.7 \\ H_{A}: \mu>63.7 \end{array} \]

Step 1 ：Define the null hypothesis \(H_{0}: \mu=63.7\) and the alternative hypothesis \(H_{A}: \mu>63.7\). The null hypothesis states that the mean height of women has not changed since 1990, while the alternative hypothesis states that the mean height has increased.

Step 2 ：Given a sample size of 45 women (n = 45), a sample mean height of 63.9 inches (x_bar = 63.9), and a sample standard deviation of 1.42 inches (s = 1.42). The mean height in 1990 was 63.7 inches (mu = 63.7).

Step 3 ：Calculate the t-statistic using the formula: \(t = \frac{x_{bar} - \mu}{s / \sqrt{n}}\). Substituting the given values, the t-statistic is approximately 0.945.

Step 4 ：Calculate the p-value, which is the probability of obtaining a result as extreme as the observed data, assuming the null hypothesis is true. The p-value is approximately 0.175.

Step 5 ：Compare the p-value with the standard significance level of 0.05. If the p-value is less than 0.05, reject the null hypothesis. If the p-value is greater than 0.05, do not reject the null hypothesis.

Step 6 ：Since the p-value (0.175) is greater than the standard significance level (0.05), we do not have enough evidence to reject the null hypothesis.

Step 7 ：Based on this sample, we cannot conclude that the mean height of women has increased from 1990 to today. Therefore, the final answer is \(\boxed{H_{0}: \mu=63.7}\)