91Ó°ÊÓ

In the following data pairs, \(A\) represents birth rate and \(B\) represents death rate per 1000 resident population. The data are paired by counties in the Midwest. A random sample of 16 counties gave the following information. (Reference: County and City Data Book, U.S. Department of Commerce.) \(\begin{array}{l|cccccccc} \hline \text { A: } & 12.7 & 13.4 & 12.8 & 12.1 & 11.6 & 11.1 & 14.2 & 15.1 \\\ \hline B: & 9.8 & 14.5 & 10.7 & 14.2 & 13.0 & 12.9 & 10.9 & 10.0 \\ \hline \\ \hline A: & 12.5 & 12.3 & 13.1 & 15.8 & 10.3 & 12.7 & 11.1 & 15.7 \\ \hline B: & 14.1 & 13.6 & 9.1 & 10.2 & 17.9 & 11.8 & 7.0 & 9.2 \\ \hline \end{array}\) Do the data indicate a difference (either way) between population average birth rate and death rate in this region? Use \(\alpha=0.01\).

Step by step solution

01

Calculate the difference for each pair

For each county, calculate the difference between the birth rate (A) and the death rate (B). Let each difference be denoted as \( d_i = A_i - B_i \). Compute these differences for all 16 pairs.

02

Compute the mean and standard deviation of the differences

First, calculate the mean of the differences, \( \overline{d} \), by summing up all the differences and dividing by the number of counties, which is 16. Next, calculate the standard deviation \( s_d \) of the differences.

03

Formulate the null and alternative hypotheses

The null hypothesis \( H_0 \) is that there is no difference between the population average birth rate and death rate, i.e., the mean of the differences is zero, \( \mu_d = 0 \). The alternative hypothesis \( H_a \) is that there is a difference, i.e., \( \mu_d eq 0 \).

04

Calculate the test statistic

Use the formula for the test statistic: \( t = \frac{\overline{d}}{\frac{s_d}{\sqrt{n}}} \), where \( n \) is the number of data pairs, which is 16. Compute the value of \( t \) using the mean and standard deviation of the differences calculated earlier.

05

Determine the critical value and decision rule

For \( \alpha = 0.01 \) and 15 degrees of freedom (since \( n - 1 = 15 \)), find the critical \( t \) value from the t-distribution table for a two-tailed test. The decision rule is to reject the null hypothesis \( H_0 \) if the absolute value of the calculated \( t \) statistic is greater than the critical \( t \) value.

06

Make a conclusion

Compare the calculated \( t \) value to the critical \( t \) value. If \(|t|\) is greater, reject \( H_0 \), indicating there is a significant difference. Otherwise, do not reject \( H_0 \), indicating no significant difference.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Paired t-test

The paired t-test is a statistical method used to compare two related samples, often the same group twice under different conditions or at different times. In simple terms, it assesses whether the mean difference between paired observations is significantly different from zero.

• This type of test is particularly useful when we are dealing with pairs of data, like in the situation of birth rates versus death rates across various counties.
• It's important to note that the pairs are naturally matched, such as before-and-after measurements or, in this case, different yet related measurements from the same counties.

For our specific exercise with birth and death rates, we need to calculate the differences for each pair and then carry out the test to see if these differences, on average, differ significantly from zero. This helps determine if there is a real difference in rates across the population being studied.

Birth Rate vs Death Rate

Birth rate refers to the number of live births in a given year per 1,000 people in the population. On the other hand, the death rate indicates how many deaths occur per 1,000 people in the same timeframe. These rates are crucial in assessing population growth and health in a given area.

• In our exercise, we are comparing these two metrics across counties to see if there is a statistically significant difference in their averages.
• Understanding the balance between birth and death rates can offer insights into population dynamics, policy effectiveness, and future societal planning needs.

By analyzing the paired differences of these rates, we can explore the extent of any imbalance, guiding policy and intervention strategies.

t-distribution

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. This means it is more prone to producing values that fall far from its mean. It is particularly useful when dealing with small sample sizes.

• In a paired t-test, the t-distribution helps us determine the critical value of the t-statistic for our hypothesis test.
• This distribution becomes crucial when we have limited data, as it accounts for the uncertainty in the estimate of the standard deviation.

For our example, with a sample size of 16 counties, we use the t-distribution to find the critical value, helping us decide whether to reject or not reject the null hypothesis.

Two-tailed Test

A two-tailed test in statistical hypothesis testing checks for the possibility of an effect in two directions. In other words, it tests for differences that are either higher or lower than zero.

• This is appropriate when we are unsure of what direction the difference might be, as is the case with birth vs. death rates.
• Using a two-tailed test means we are assessing both the possibility that the birth rate could be significantly higher than the death rate, or vice versa.

In our exercise, using a two-tailed test involves considering any difference, not just one that is in a specific direction, which is key to thoroughly understanding these rates' comparative nature.