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3.124 Donating Blood to Grandma? There is some evidence that "young blood" might improve the health, both physically and cognitively, of elderly people (or mice). Exercise 2.69 on page 75 introduces one study in which old mice were randomly assigned to receive transfusions of blood from either young mice or old mice. Researchers then measured the number of minutes each of the old mice was able to run on a treadmill. The data are stored in YoungBlood. We wish to estimate the difference in the mean length of time on the treadmill, between those mice getting young blood and those mice getting old blood. Use StatKey or other technology to find and interpret a \(90 \%\) confidence interval for the difference in means.

Step by step solution

01

Gather Data

Firstly, gather the data from the YoungBlood study. You need two sets of data: first, the length of time old mice that received young blood were able to run on a treadmill. Second, the length of time old mice that received old blood were able to run on a treadmill.

02

Calculate Point Estimates

Now calculate the means of the two groups, which serve as the point estimates. Use the formula for the mean: \(\bar{X} = \frac{1}{N} \sum_{i=1}^{N} X_i\). Calculate this for both data sets.

03

Calculate Standard Errors

Next, calculate the standard errors for both groups. The standard error is calculated as \(\frac{S}{\sqrt{n}}\), where \(S\) is the standard deviation and \(n\) is the number of observations. Calculate this for both data sets.

04

Calculate Degrees of Freedom

Degrees of freedom can be calculated using the formula: \(df = n1 + n2 - 2\), where \(n1\) and \(n2\) are the sample sizes of both groups.

05

Define Confidence Interval

We're interested in a 90% confidence interval. From a t-table, find the t-value that corresponds to the 90% confidence and your degree of freedom.

06

Calculate Confidence Interval

Using your obtained t value, you calculate the confidence interval using the formula: \((\overline{x_1} - \overline{x_2}) ± t \sqrt{s_1^2/n_1 + s_2^2/n_2}\). Here, \(\overline{x_1}\) and \(\overline{x_2}\) are the means, \(s_1\) and \(s_2\) are the standard deviations, \(n_1\) and \(n_2\) are the sample sizes of two groups, and \(t\) is the t-value from the t-table.

07

Interpret Result

The resulting interval forms your 90% confidence interval for the difference in population means. If this interval does not include zero, we can conclude that the two population means are statistically different at the 90% confidence level.

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Key Concepts

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

Understanding the Difference in Means

When we talk about the difference in means, we're interested in finding how much one group's average differs from another's. In our young blood study, this means comparing the running times of old mice that received young blood versus those with old blood. This helps us understand whether the type of blood affects endurance.

• Calculate the average running time for each group, known as the point estimate. Use the formula: \(\bar{X} = \frac{1}{N} \sum_{i=1}^{N} X_i\).
• Subtract one mean from the other to find the difference in means.

This difference in means is crucial for determining if there's a significant effect or if the variations are just due to chance.

Calculating the Standard Error

The standard error tells us how much the sample mean is expected to vary from the true population mean. It's a measure of the sample's accuracy.

• Calculate using \(\frac{S}{\sqrt{n}}\), where \(S\) is the standard deviation and \(n\) is the sample size.
• The smaller the standard error, the closer our sample mean is likely to be to the true population mean.

Understanding standard error helps us gauge the reliability of our mean difference in the context of variability.

Exploring Degrees of Freedom

Degrees of freedom play a big role when determining the correct t-value for confidence intervals. They refer to the number of values in a calculation that are free to vary.

• Calculated as \(df = n1 + n2 - 2\), where \(n1\) and \(n2\) are the sample sizes of each group.
• The degrees of freedom affect how the distribution looks and thus influences the t-value you use.

By knowing the degrees of freedom, you ensure that you're using the correct distribution to make accurate predictions about the confidence interval.