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The degenerative disease osteoarthritis most frequently affects weight-bearing joints such as the knee. The article 'Evidence of Mechanical Load Redistribution at the Knee Joint in the Elderly When Ascending Stairs and Ramps" (Annals of Biomed. Engr." 2008: 467-476) presented the following summary data on stance duration (ms) for samples of both older and younger adults. $$ \begin{array}{lccc} \text { Age } & \text { Sample Size } & \text { Sample Mean } & \text { Sample SD } \\ \hline \text { Older } & 28 & 801 & 117 \\ \text { Younger } & 16 & 780 & 72 \\ \hline \end{array} $$ Assume that both stance duration distributions are normal. a. Calculate and interpret a \(99 \% \mathrm{CI}\) for true average stance duration among elderly individuals. b. Carry out a test of hypotheses at significance level \(.05\) to decide whether true average stance duration is larger among elderly individuals than among younger individuals.

Step by step solution

01

Calculate the standard error for the older group

The standard error (SE) for older adults is calculated using the formula for standard deviation: \( SE_{older} = \frac{SD_{older}}{\sqrt{n}} \), where \( SD_{older} \) is the sample standard deviation (117) and \( n \) is the sample size (28). Compute \( SE_{older} = \frac{117}{\sqrt{28}} \approx 22.09 \).

02

Determine the critical t-value for a 99% confidence interval

For a 99% confidence interval and 27 degrees of freedom (\( n-1 = 28-1 \)), use the t-distribution table to find the critical t-value. Looking up a 99% confidence interval, we find \( t^* = 2.771 \).

03

Calculate the 99% confidence interval for the older group

The confidence interval for the true average is given by \( \bar{x} \pm t^* \times SE \). For older adults, use \( 801 \pm 2.771 \times 22.09 \), giving CI \( 801 \pm 61.19 \), which results in the interval (739.81, 862.19).

04

Set up the hypothesis test

Identify the null and alternative hypotheses. Null hypothesis \( H_0: \mu_{older} = \mu_{younger} \) and alternative hypothesis \( H_a: \mu_{older} > \mu_{younger} \). We want to determine if the stance duration is significantly larger among older than younger individuals.

05

Calculate the standard error for the difference in means

The standard error for the difference in means is \( SE_{diff} = \sqrt{\frac{SD_{older}^2}{n_{older}} + \frac{SD_{younger}^2}{n_{younger}}} \), which is \( \sqrt{\frac{117^2}{28} + \frac{72^2}{16}} \approx 24.81 \).

06

Calculate the test statistic

Use the test statistic formula for comparing two means: \( t = \frac{\bar{x}_{older} - \bar{x}_{younger}}{SE_{diff}} = \frac{801 - 780}{24.81} \approx 0.846 \).

07

Determine the decision for the hypothesis test

Find the critical t-value from the t-distribution table for \( 42 \) degrees of freedom at \( \alpha = 0.05 \). With a critical value of \( t^* = 1.684 \), compare it with the calculated t (0.846). Since 0.846 < 1.684, we fail to reject \( H_0 \).

08

Interpret the results

The 99% confidence interval for older individuals’ stance duration is (739.81, 862.19), indicating the range where the true average likely lies. The hypothesis test suggests no significant difference at the 0.05 level in average stance duration between older and younger individuals.

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Key Concepts

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

Confidence Intervals

When estimating a population parameter like the true average stance duration among older adults, confidence intervals provide a range of plausible values. A confidence interval quantifies the uncertainty around the sample mean by considering variability within the sample data.
In our exercise, we determined a 99% confidence interval for the older group's average stance duration. This interval tells us that, with 99% confidence, the true average lies between 739.81 ms and 862.19 ms.
The confidence level indicates how sure we are that the interval includes the true mean. A 99% confidence level offers more certainty compared to, say, a 95% confidence level, but it results in a wider range.

• A higher confidence level means a wider interval, implying more uncertainty but greater assurance that the interval covers the true parameter.
• Reducing the confidence level narrows the interval, which means less assurance about capturing the true parameter.

Interpreting confidence intervals requires understanding these trade-offs between precision and certainty.

Standard Error

The standard error (SE) reflects sampling variability and is crucial for building confidence intervals and hypothesis testing. It estimates the standard deviation of the sample mean distribution.
For the older group, the standard error was calculated using the formula: \[ SE = \frac{SD}{\sqrt{n}} \]
where SD is the sample standard deviation and n is the sample size. For older individuals, the SE calculated is approximately 22.09 ms.
The standard error gives us an idea of how much the sample mean is expected to vary from the actual population mean. A smaller SE indicates less variance, suggesting our sample mean is close to the true mean.

• A large sample size typically results in a smaller SE, increasing precision.
• Smaller SD also reduces SE, enhancing the reliability of the sample mean.

Thus, the standard error is a pivotal component in statistical analysis, allowing accurate interpretation of sample data.

T-Distribution

The T-distribution is a crucial concept when dealing with small sample sizes or unknown population standard deviations. It resembles the normal distribution but has heavier tails, allowing for variability in small datasets.
In hypothesis testing and confidence interval estimation, we often rely on the T-distribution rather than the normal distribution to account for uncertainties in smaller samples.
For our exercise, the critical t-value for a 99% confidence interval with 27 degrees of freedom (n-1 for the older group) was found using this distribution, yielding a t-value of 2.771.
The T-distribution becomes nearly identical to the normal distribution as sample size increases. Therefore:

• With smaller samples (n < 30), the T-distribution is preferred for its ability to manage uncertainty effectively.
• As the sample size grows, using a normal distribution becomes more appropriate due to the central limit theorem.

Understanding the T-distribution enhances the robustness of statistical analyses across varied data conditions.

Sample Mean

The sample mean is at the heart of statistical estimates, providing a basis for constructing confidence intervals and conducting hypothesis tests.
In this exercise, the sample mean for the older group was 801 ms, serving as a point estimate of the population mean.
The sample mean summarizes the central tendency of a dataset and, when combined with measures like standard error, helps in drawing conclusions about the population.
When comparing two groups, like in our hypothesis test, the difference in sample means indicates potential variations in population parameters. In this example:

• The difference between older and younger groups' means (801 ms and 780 ms) helps pose or test hypotheses about demographic differences.
• The comparison of sample means, coupled with standard errors, guides us to understand if the observed differences are statistically significant.

Thus, the sample mean is fundamental in our journey from data analysis to meaningful interpretation.