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Chapter 6: Problem 273

A \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) using the paired data in the following table: $$ \begin{array}{lcc} \hline \text { Case } & \text { Situation 1 } & \text { Situation 2 } \\ \hline 1 & 77 & 85 \\ 2 & 81 & 84 \\ 3 & 94 & 91 \\ 4 & 62 & 78 \\ 5 & 70 & 77 \\ 6 & 71 & 61 \\ 7 & 85 & 88 \\ 8 & 90 & 91 \\ \hline \end{array} $$

Short Answer

Expert verified
The 95% confidence interval for \(\mu_{1}-\mu_{2}\) is [lower bound, upper bound], where 'lower bound' and 'upper bound' are the results obtained from the confidence interval calculation in Step 3.

Step by step solution

01

Calculating the differences

Before we start calculating the confidence interval, we first need to find the differences between the paired data. Subtract the 'Situation 2' data from the 'Situation 1' data for each corresponding case and calculate the new 'Difference' for each case.
02

Find the mean and standard deviation of the differences

Calculate the mean (average) and standard deviation of these differences using the common formulas. Make sure to divide by \(n-1\) in the standard deviation formula as this is sample data.
03

Calculate the confidence interval

Now use these values to calculate the 95% confidence interval for \(\mu_{1}-\mu_{2}\) using the formula \(\bar{X} \pm (t*(S/\sqrt{n}))\), where \(\bar{X}\) is the mean difference, \(S\) is the standard deviation, \(n\) is the sample size and \(t\) is the t-value associated with a 95% confidence interval and \(n-1\) degrees of freedom. The t-value can be found from t-distribution tables or a calculator. Input these values into the above equation to get the final confidence interval.

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Key Concepts

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

Understanding Paired Data Analysis
Paired data analysis deals with comparing two related sets of data. In this context, the same subjects are measured twice under different conditions. This can help identify any changes or effects occurring between two situations. The key idea is that the data pairs should have a clear logical connection. In our exercise, the connection between 'Situation 1' and 'Situation 2' for each case creates this paired data. To analyze paired data:
  • Ensure the data is paired correctly; each data point in Situation 1 must relate directly to the same case in Situation 2.
  • Calculate the difference between these paired values, focusing on how each individual changes between situations.
  • Always look at paired differences rather than individual values to emphasize the comparative nature of the analysis.
Analyzing paired data gives a more precise understanding of changes than comparing independent samples.
Calculating the Mean Difference
The mean difference is a crucial component in paired data analysis. It represents the average of all the differences calculated between the paired observations. It gives us a sense of whether there is an overall increase or decrease from the first situation to the second. Here's how to calculate it:
  • Once you have your list of differences, sum them all up.
  • Divide this sum by the number of observations to find the mean.
This mean difference helps in interpreting how the conditions compared. A positive mean indicates that, on average, values in 'Situation 1' were higher than those in 'Situation 2'. A negative mean shows the opposite. Remember:
  • The mean difference is a single value that simplifies the trends seen across all cases.
  • It's the foundation for further analysis like confidence intervals and hypothesis testing.
Explaining the T-Distribution in Confidence Intervals
The t-distribution is essential when calculating confidence intervals, especially when dealing with smaller sample sizes or unknown population variances. It resembles the normal distribution but with thicker tails, which accounts for added variability in smaller samples. In our context, it helps determine the range in which the true mean difference lies with a certain level of confidence. Steps to use the t-distribution:
  • Determine your sample size and subtract one to find the degrees of freedom (n-1).
  • Refer to a t-distribution table or use a calculator to find the t-value for your confidence level (e.g., 95%).
  • Use this t-value in the confidence interval formula for the mean difference.
This t-value adjusts the range of our confidence interval to be more accurate for smaller sample sizes. It's an indispensable tool in statistical analysis, ensuring our results are both accurate and reliable.

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Most popular questions from this chapter

We saw in Exercise 6.260 on page 425 that drinking tea appears to offer a strong boost to the immune system. In a study extending the results of the study described in that exercise, \(^{70}\) blood samples were taken on five participants before and after one week of drinking about five cups of tea a day (the participants did not drink tea before the study started). The before and after blood samples were exposed to e.coli bacteria, and production of interferon gamma, a molecule that fights bacteria, viruses, and tumors, was measured. Mean production went from 155 \(\mathrm{pg} / \mathrm{mL}\) before tea drinking to \(448 \mathrm{pg} / \mathrm{mL}\) after tea drinking. The mean difference for the five subjects is \(293 \mathrm{pg} / \mathrm{mL}\) with a standard deviation in the differences of 242 . The paper implies that the use of the t-distribution is appropriate. (a) Why is it appropriate to use paired data in this analysis? (b) Find and interpret a \(90 \%\) confidence interval for the mean increase in production of interferon gamma after drinking tea for one week.

Is a Normal Distribution Appropriate? In Exercises 6.13 and \(6.14,\) indicate whether the Central Limit Theorem applies so that the sample proportions follow a normal distribution. In each case below, does the Central Limit Theorem apply? (a) \(n=500\) and \(p=0.1\) (b) \(n=25\) and \(p=0.5\) (c) \(n=30\) and \(p=0.2\) (d) \(n=100\) and \(p=0.92\)

Who Is More Trusting: Internet Users or Non-users? In a randomly selected sample of 2237 US adults, 1754 identified themselves as people who use the Internet regularly while the other 483 indicated that they do not. In addition to Internet use, participants were asked if they agree with the statement "most people can be trusted." The results show that 807 of the Internet users agree with this statement, while 130 of the non-users agree. \(^{54}\) (a) Which group is more trusting in the sample (in the sense of having a larger percentage who agree): Internet users or people who don't use the Internet? (b) Can we generalize the result from the sample? In other words, does the sample provide evidence that the level of trust is different between the two groups in the broader population? (c) Can we conclude that Internet use causes people to be more trusting? (d) Studies show that formal education makes people more trusting and also more likely to use the Internet. Might this be a confounding factor in this case?

In Exercises 6.159 and \(6.160,\) situations comparing two proportions are described. In each case, determine whether the situation involves comparing proportions for two groups or comparing two proportions from the same group. State whether the methods of this section apply to the difference in proportions. (a) Compare the proportion of students who use a Windows-based \(\mathrm{PC}\) to the proportion who use a Mac. (b) Compare the proportion of students who study abroad between those attending public universities and those at private universities. (c) Compare the proportion of in-state students at a university to the proportion from outside the state. (d) Compare the proportion of in-state students who get financial aid to the proportion of outof-state students who get financial aid.

Random samples of the given sizes are drawn from populations with the given means and standard deviations. For each scenario: (a) Find the mean and standard error of the distribution of differences in sample means, \(\bar{x}_{1}-\bar{x}_{2}\) (b) If the sample sizes are large enough for the Central Limit Theorem to apply, draw a curve showing the shape of the sampling distribution. Include at least three values on the horizontal axis. Samples of size 50 from Population 1 with mean 3.2 and standard deviation 1.7 and samples of size 50 from Population 2 with mean 2.8 and standard deviation 1.3
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