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Chapter 11: Problem 11

What is the typical age difference between husband and wife? The following data represent the ages of husbands and wives, based on results from the Current Population Survey. (a) What is the response variable in this study? (b) Is the sampling method dependent or independent? Explain. (c) Use the data to estimate the mean difference in age of husband and wives with \(95 \%\) confidence. Explain the technique that you used.

Short Answer

Expert verified
(a) The typical age difference is the response variable. (b) The sampling method is dependent. (c) Estimate the mean difference using paired sample t-test formula with 95% confidence.

Step by step solution

01

- Identify the Response Variable

The response variable is the variable that the study aims to measure or predict. In this case, the study aims to measure the ages of husbands and wives. The typical age difference, therefore, is the response variable. It answers the question: 'What is the typical age difference between husband and wife?'.
02

- Determine if the Sampling Method is Dependent or Independent

Dependent sampling methods are used when there is a natural pairing between the samples, such as when each data point in one sample corresponds to one in another. Here, the ages of husbands and wives are naturally paired since each husband's age is paired with his wife's age. Hence, the sampling method is dependent.
03

- Estimate the Mean Difference with 95% Confidence Level

To estimate the mean difference in age with 95% confidence, follow these steps:1. Compute the differences between each husband's age and his wife's age.2. Calculate the mean and standard deviation of these differences.3. Use the formula for the confidence interval of the mean difference in dependent samples:\[ \bar{d} \text{ 卤 } t_{\frac{伪}{2}} \frac{s_d}{\frac{\text{鈭歯}}} \text{, where:} \]\(\bar{d}\) = mean of the differences,\(t_{\frac{伪}{2}}\) = critical value from the t-distribution for 95% confidence,\(s_d\) = standard deviation of the differences,\(n\) = number of pairs.Calculate these values to get the confidence interval for the mean age difference.

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Key Concepts

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

response variable
In statistical studies, the response variable is essentially the main focus of the research. It is what the study seeks to measure or predict.
In our study, we are particularly interested in understanding the 'typical age difference between husbands and wives'.
This difference is what we call the response variable, as it directly answers the core question posed by the study.

For clarity, response variables can sometimes be referred to as 'dependent variables' because their values depend on other factors.
In our case, the actual ages of the husbands and wives are these factors.
dependent sampling method
When conducting studies, the way we collect samples plays a crucial role in how we analyze data later on.
One common method is the dependent sampling method. This method occurs when each data point in one sample is paired with a data point in another sample.

In our study, every husband's age is naturally paired with his respective wife's age.
This natural pairing means that the samples are dependent.
  • Dependence here implies that knowing one husband's age gives direct context to his wife's age.
  • This information helps us make more accurate comparisons and predictions.
Therefore, our study employs a dependent sampling method.
mean difference
The mean difference is a fundamental concept in statistics, especially when comparing two related samples.
It is the average of the differences between paired observations.

For our study, to compute the mean difference between the ages of husbands and wives:
  • Subtract each wife's age from her husband's age.
  • Sum all those differences.
  • Divide by the total number of pairs to get the mean difference.
This value provides a central estimate of the typical age gap, giving us a clear idea about the typical age difference in couples.
95% confidence interval
A confidence interval gives a range within which we expect the true mean difference to lie, with a certain level of confidence.
In our case, we want a 95% confidence interval.
This means we can be 95% confident that the interval contains the true mean age difference between husbands and wives.

To calculate it in the context of our study:
  • First, determine the mean difference and its standard deviation.
  • Use the formula: \(\bar{d} \text{±} t_{\frac{伪}{2}} \frac{s_d}{\frac{\text{鈭歯}}}\)
  • Here, \(\bar{d}\) is the mean difference, \(t_{\frac{伪}{2}}\) is the critical value from the t-distribution (for 95% confidence), \(s_d\) is the standard deviation of the differences, and \(n\) is the number of pairs.
This formula will provide the range within which the true mean difference lies.
t-distribution
The t-distribution is a statistical distribution particularly useful when the sample size is small.
Unlike the more known normal distribution, the t-distribution has heavier tails, meaning it is more prone to producing values that fall far from its mean.
This characteristic helps make more accurate inferences with smaller sample sizes.

When calculating the confidence interval for our mean difference, we use the \(t\)-distribution:
  • It provides the critical value (\(t_{\frac{伪}{2}}\)) needed in our confidence interval calculation.
  • This value adjusts our confidence interval to better reflect the reality when dealing with a small sample size.
  • So, in our study of age differences between husbands and wives, using the \(t\)-distribution is crucial for accurate estimation.

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Most popular questions from this chapter

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