Question

Research has shown that for mammals giving birth to a son versus a daughter places a greater strain on mothers. Does this affect the health of their next child? A study compared the birth-weights of humans born after a male versus after a female. Summary statistics for the sample of size 76 are given in the following table; the data were normally distributed. \({ }^{40}\) Consider a \(t\) test, with a directional alternative, to investigate the research hypothesis that birth weight is lower when the elder sib ling is male. (a) Write the null hypothesis and the alternative hypothesis in symbols. (b) Here is computer output for the \(t\) test. Explain what the \(P\) -value means in the context of this study. $$ \mathrm{t}=-2.145, \mathrm{df}=69.54, \mathrm{p}-\mathrm{va} 7 \mathrm{ue}=0.0177 $$ alternative hypothesis: true difference in means is less than 0 (c) If \(\alpha=0.05,\) what is your conclusion regarding \(H_{0} ?\)

Short answer

H_0: \(\mu_1 = \mu_2\). H_a: \(\mu_1 < \mu_2\). P-value of 0.0177 indicates that there is only a 1.77% chance of such results if the null hypothesis were true. With \(\alpha = 0.05\), we reject H_0, supporting the hypothesis that birth weight is lower when the elder sibling is male.

Step-by-step solution

Formulate the Hypotheses

The null hypothesis (H_0) assumes there is no difference in the birth weights of children born after a male sibling compared to those born after a female sibling. In symbols, this can be stated as H_0: \(\mu_1 = \mu_2\), where \(\mu_1\) is the mean birth weight after a male sibling and \(\mu_2\) is the mean birth weight after female sibling. The alternative hypothesis (H_a) is that the birth weight is lower when the elder sibling is male, stated as H_a: \(\mu_1 < \mu_2\).

Interpret the P-value

The P-value of 0.0177 represents the probability of observing a test statistic as extreme as t = -2.145, or more extreme, given that the null hypothesis is true. Since the P-value is a measure of evidence against the null hypothesis, a lower P-value indicates more evidence against H_0. In the context of this study, a P-value of 0.0177 suggests that there is a low probability (1.77%) of observing a difference in birth weights as large as the one in the sample data, if in reality there was no difference between the groups (if H_0 were true).

Draw a Conclusion

Given that \(\alpha = 0.05\), we compare the P-value to \(\alpha\). Since the P-value (0.0177) is less than \(\alpha\) (0.05), we reject the null hypothesis H_0. This means that there is sufficient evidence at the 0.05 significance level to support the research hypothesis that the birth weight is lower when the previous child is a male.

Key concepts

Null Hypothesis

The null hypothesis, often symbolized as \(H_0\), is a cornerstone of statistical testing, serving as a default statement to be tested and potentially rejected. It posits no effect or no difference in the case of biostatistics studies. For the exercise in question, the null hypothesis states that the mean birth weight of children born after a male sibling (\(\mu_1\)) and after a female sibling (\(\mu_2\)) is the same, which can be formally written as \(H_0: \mu_1 = \mu_2\). It is articulated from a stance of skepticism, proposing that any observed difference in birth weights is due to chance rather than a particular cause (in this case, the gender of the preceding sibling).

The merit of the null hypothesis lies in its role as a starting point. It provides a baseline measure that the data under scrutiny can be compared against. Our challenge, through statistical testing, is to determine whether the observed data provide enough evidence to reject this null hypothesis in favor of an alternative explanation.

Alternative Hypothesis

Contrasting the null hypothesis is the alternative hypothesis, noted as \(H_a\) or \(H_1\), which directly challenges the null by proposing a specific direction of effect or relationship. In our exercise, the alternative hypothesis declares that the birth weight of a child born after a male sibling (\(\mu_1\)) is less than that of a child born after a female sibling (\(\mu_2\)), expressed as \(H_a: \mu_1 < \mu_2\).

This directional hypothesis is significant for two reasons. Firstly, it aligns with the specific research question regarding the impact of sibling gender on subsequent birth weight. Secondly, it dictates the nature of the statistical test to be used—in this case, a one-tailed \(t\)-test that assesses whether the birth weights following a male sibling are statistically lower. Should evidence refute the null hypothesis, the alternative hypothesis emerges as the more plausible explanation in light of the available data.

P-value Interpretation

The P-value, an abbreviation for the probability value, is a critical component within the realm of hypothesis testing and is essential for interpreting study results. It provides a numerical measure to help us determine the strength of the evidence against the null hypothesis. A P-value indicates the probability of obtaining results at least as extreme as those observed during the experiment, assuming that the null hypothesis is true.

In the context of the exercise, a P-value of 0.0177 implies a probability of 1.77% that the data would show an equally large or larger difference in birth weights due to random chance alone, if there truly were no difference based on the sibling's gender. It's a way of quantifying the likelihood that the variation observed is purely accidental; hence, a smaller P-value suggests it's less likely to be due to random chance and more likely indicative of a real effect. The interpretation of the P-value grounds our decision on whether to accept the null hypothesis or consider the alternative hypothesis more favorable based on the evidence.

Statistical Significance

Statistical significance is the judgment we make about the strength of the evidence against the null hypothesis, given the P-value in relation to an agreed-upon threshold called the \(\alpha\)-level. Typically, the \(\alpha\)-level is set at 0.05 (5%), although it can be adjusted depending on the field of study or under specific considerations concerning the consequences of making an error in judgment.

In the exercise under review, the computed P-value of 0.0177 is below the \(\alpha\)-level of 0.05. This suggests that the observed data is unlikely to occur by chance alone and grants enough empirical weight to reject the null hypothesis. As a result, one may confidently infer that the data offers substantial evidence to uphold the alternative hypothesis—that there is indeed a statistical significance in the lower birth weights following a male sibling compared to a female sibling within this study. This conclusion carries weight because it surpasses the set standard for signaling a meaningful difference rather than a random occurrence.