Problem 6 - An Introduction to Mathematical Statistics and Its Applications

Question

Ring Lardner was one of this country's most popular writers during the \(1920 \mathrm{~s}\) and \(1930 \mathrm{~s}\). He was also a chronic alcoholic who died prematurely at the age of forty-eight. The following table lists the life spans of some of Lardner's contemporaries (39). Those in the sample on the left were all problem drinkers; they died, on the average, at age sixty-five. The twelve (sober) writers on the right tended to live a full ten years longer. Can it be argued that an increase of that magnitude is statistically significant? Test an appropriate null hypothesis against a one-sided \(H_{1}\). Use the \(0.05\) level of significance. (Note: The pooled sample standard deviation for these two samples is 13.9.)

Step-by-Step Solution

Answer

Reject or fail to reject the null hypothesis based on the comparison of the test statistic and critical value. If rejected, it can be said that the increase in lifespan between problem drinkers and sober writers is statistically significant at a 0.05 level of significance.

Step 1: Formulate the hypotheses

The null hypothesis \(H_{0}\) states there is no significant difference in lifespan between the two groups, that is, the mean difference is zero. The alternate hypothesis \(H_{1}\), which is one-sided in this case, states that the difference in means is greater than zero. Mathematically these can be denoted as: \(H_{0}: \mu_{1} - \mu_{2} = 0 \) and \(H_{1}: \mu_{1} - \mu_{2} > 0\), where \(\mu_{1}\) represents the average lifespan of problem drinkers and \(\mu_{2}\) of sober individuals.

Step 2: Calculate the test statistic

The test statistic for this exercise is the difference in sample means divided by the standard deviation of the difference. Given that problem drinkers have an average lifespan of 65 years, sober individuals an average lifespan of 75 years, and the pooled standard deviation is 13.9, the test statistic can be calculated as \((65-75)/13.9\).

Step 3: Compare the test statistic with the critical value

For a one-sided test at a 0.05 level of significance, the critical value from z-table is approximately 1.645 (since 1-0.05 equates to the 0.95 percentile in the z-table). If the calculated test statistic is greater than this critical value, we reject the null hypothesis. Conversely, if the test statistic is less than the critical value, we fail to reject the null hypothesis.

Step 4: Make the decision

After comparing the calculated test statistic with the critical value, conclude whether or not to reject the null hypothesis. If it is rejected, it indicates that the difference in lifespans between problem drinkers and sober individuals is statistically significant.

Key Concepts

Null Hypothesis

The null hypothesis, symbolically represented as \(H_{0}\), is a fundamental notion in statistical significance testing. It constitutes the default assumption that there is no effect or no difference between two or more groups or variables.

In the context of the exercised problem involving Ring Lardner and his contemporaries, the null hypothesis posits that the average lifespan (\(\mu_{1}\)) of problem drinkers and the average lifespan (\(\mu_{2}\)) of sober writers do not differ, which is mathematically expressed as \(H_{0}: \mu_{1} - \mu_{2} = 0\). This hypothesis serves as the starting point for testing. It is the skepticism we aim to either dismiss or fail to disprove, based on the evidence provided by our test statistic. If we find sufficient evidence to refute the null hypothesis, we may lend support to the alternative hypothesis, which suggests there is a meaningful difference to be considered—in this case, a difference in lifespan related to alcohol consumption.

Test Statistic Calculation

The test statistic is a standardized value that allows us to make decisions about the null hypothesis. It's calculated using sample data and helps us determine the likelihood of observing a result as extreme as, or more extreme than, the one in our sample if the null hypothesis were true.

For our exercise, the test statistic is determined by computing the difference in sample means—65 for problem drinkers and 75 for sober individuals—and dividing it by the pooled standard deviation of 13.9. This results in a test statistic of \( (65 - 75) / 13.9 \) which simplifies to approximately -0.72. This calculated value would then be compared to a critical value from a statistical table corresponding to the chosen significance level, typically denoted as \( \alpha \), which in this case is 0.05.

Difference in Means

The difference in means is a comparison between the average values of two different groups or conditions, which, for this scenario, are the lifespans of problem drinkers versus sober writers. It is an essential part of the analysis as we are trying to assess if the observed difference in average lifespan is statistically significant or could have occurred by random chance.

In statistical testing, we measure the magnitude of this difference alongside its variability using the standard deviation. We then normalize this value to obtain a test statistic that can be judged against a critical value, facilitating our decision-making process about the null hypothesis. This comparison aims to determine if the empirical evidence suggests a meaningful difference in lifespans derived from alcohol consumption habits.