Data released by the National Center for Health Statistics...

Data released by the National Center for Health Statistics showed that the mean age at which women had their first child was 25.0 in 2006 (The Wall Street Journal, February 4 , 2009 ). The reporter, Sue Shellenbarger, noted that this was the first decrease in the average age at which women had their first child in several years. A recent sample of 42 women provided the data in the website file named FirstB irth concerning the age at which these women had their first child. Do the data indicate a change from 2006 in the mean age at which women had their first child? Use $a=.05$.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population parameter based on sample data. It involves comparing an observed statistic to what would be expected under a stated hypothesis about the population. The process is designed to assess whether the evidence in the sample is strong enough to reject a null hypothesis in favor of an alternative hypothesis.

Null and Alternative Hypotheses

The null hypothesis (often denoted as H?) represents a statement of no effect or no difference; in this context, it would assert that there is no change in the mean age at which women have their first child compared to a known value. The alternative hypothesis (H? or Ha) represents a statement that contradicts the null, such as a change—in either direction—in the mean age. These hypotheses provide the framework for the statistical test.

Significance Level

A significance level, such as 0.05, is a threshold set by the researcher to determine the risk of incorrectly rejecting the null hypothesis (Type I error). It defines the probability of making this error, ensuring that the results are statistically significant if the observed effects are unlikely to have occurred by random chance alone.

Two-Tailed Test

A two-tailed test is used when deviations in both directions (either higher or lower) from the hypothesized value are considered significant. It checks for any significant difference from the known mean, rather than testing for an increase or a decrease exclusively, which is appropriate when there is interest in any change in the parameter.

Test Statistic

The test statistic, such as a t-statistic or z-statistic, measures how far the sample statistic deviates from the null hypothesis value, in units of the standard error of the sample statistic. The choice of test statistic generally depends on factors such as sample size and whether the population standard deviation is known or estimated from the sample.

p-value

The p-value quantifies the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is correct. It serves as a tool to compare against the significance level; if the p-value is less than the significance level, the null hypothesis is rejected, indicating a statistically significant result.

Central Limit Theorem

The Central Limit Theorem is a fundamental principle stating that, given a sufficiently large sample size, the distribution of the sample mean will approximate a normal distribution regardless of the distribution of the underlying population. This theorem justifies the use of normal or t-distribution-based tests even when the original data are not normally distributed.

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