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Chapter 9: Problem 12

The accompanying summary data on total cholesterol level (mmol/l) was obtained from a sample of Asian postmenopausal women who were vegans and another sample of such women who were omnivores ("Vegetarianism, Bone Loss, and Vitamin D: A Longitudinal Study in Asian Vegans and Non-Vegans," European \(J\). of Clinical Nutr., 2012: 75-82). \begin{tabular}{lccc} Diet & Sample Size & Sample Mean & Sample SD \\ \hline Vegan & 88 & \(5.10\) & \(1.07\) \\ Omnivore & 93 & \(5.55\) & \(1.10\) \\ \hline \end{tabular} Calculate and interpret a \(99 \%\) CI for the difference between population mean total cholesterol level for vegans and population mean total cholesterol level for omnivores (the cited article included a 95\% CI). [Note: The article described a more sophisticated statistical analysis for investigating bone density loss taking into account other characteristics ("covariates") such as age, body weight, and various nutritional factors; the resulting CI included 0 , suggesting no diet effect.]

Short Answer

Expert verified
The 99% confidence interval is approximately (-0.8773, -0.0227), suggesting vegans have lower cholesterol levels.

Step by step solution

01

Identify the Problem

We need to calculate a 99% confidence interval (CI) for the difference in means between the total cholesterol levels of two populations: vegan and omnivore Asian postmenopausal women. This involves two independent samples.
02

Gather the Sample Data

From the problem, we have: - Vegan group: - Sample Size ( - Vegan group: Sample Size ( - Vegan group: Sample Size ( Vegan group: Sample Size ( - Omnivore group: - Sample Size ( - Sample Mean ( 5.55 5.55 5.55 Sample Mean: The sample mean for omnivores is 5.55 mmol/l - Sample SD ( 1.10 1.10 1.10 Sample SD: The sample standard deviation for omnivores is 1.10 mmol/l.
03

Mean Difference and Pooled Variance

The difference in sample means is calculated as follows:\[\bar{x}_1 - \bar{x}_2 = 5.10 - 5.55 = -0.45\]The pooled variance is:\[s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}\]Substituting the values:\[s_1^2 = 1.07^2, s_2^2 = 1.10^2\]\[s_p^2 = \frac{(88 - 1)(1.07^2) + (93 - 1)(1.10^2)}{88 + 93 - 2} = \frac{87 \times 1.1449 + 92 \times 1.21}{179}\]\[s_p^2 = \frac{100.2063 + 111.32}{179} = 1.18456\]Thus, \(s_p \approx \sqrt{1.18456} = 1.0884\).
04

Calculate the Standard Error

The standard error (SE) of the difference in means is given by:\[SE = \sqrt{s_p^2 \left(\frac{1}{n_1} + \frac{1}{n_2} \right)}\]Substituting the relevant values:\[SE = \sqrt{1.18456 \left(\frac{1}{88} + \frac{1}{93} \right)}\]\[SE \approx \sqrt{1.18456 \times 0.02248} \approx \sqrt{0.026639} \approx 0.1632\]
05

Determine the Critical Value

For a 99% confidence interval, the critical value (\(t^*\)) can be found using the \(t\)-distribution table with \(df = n_1 + n_2 - 2 = 179\). Approximating from the table for a confidence level of 99%, we get \(t^* \approx 2.621\).
06

Calculate the Confidence Interval

The 99% CI for the difference in population means is:\[(\bar{x}_1 - \bar{x}_2) \pm t^* \times SE\]Substituting the values:\[CI = -0.45 \pm 2.621 \times 0.1632\]\[CI \approx -0.45 \pm 0.4273\]Thus, the 99% CI is approximately \((-0.8773, -0.0227)\).
07

Interpret the Result

The 99% confidence interval for the difference between the population mean total cholesterol levels for vegans and omnivores is approximately \((-0.8773, -0.0227)\). Since the interval does not include 0, it suggests that there is a statistically significant difference in cholesterol levels between vegans and omnivores, with vegans having lower levels.

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Key Concepts

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

Statistical Analysis in Cholesterol Studies
Statistical analysis plays a crucial role when it comes to evaluating clinical data, such as cholesterol levels in different populations.
In this context, we focus on postmenopausal women belonging to both vegan and omnivore groups.
The aim of statistical analysis here is to determine whether there's a significant difference in cholesterol levels between these two dietary groups.
This involves comparing the means of the two samples, each from the different dietary groups, to infer about the larger populations they represent.

We employ concepts such as confidence intervals and pooled variance. A confidence interval (CI) provides a range of values that likely contain the true population parameter.
In this exercise, we calculated a 99% CI for the difference in means, which tells us where the true difference in cholesterol levels likely falls.
Pooled variance, on the other hand, is used when dealing with two independent samples, like in our study on vegan and omnivore groups.
It combines data variations from both samples to provide a more accurate measure of their overall variance.

This statistical analysis essentially allows researchers to determine if observed differences could occur randomly or are indeed indicative of real effects brought on by different diets.
Cholesterol Levels Comparison
Cholesterol levels can significantly vary based on various factors, including diet. In this study, we see a comparison between cholesterol levels in vegan and omnivore postmenopausal women.
Understanding such differences is critical as cholesterol levels are closely linked to heart health and other diseases.
On average, the omnivore group showed a higher mean cholesterol level (5.55 mmol/l) compared to the vegan group (5.10 mmol/l).

To compare these groups effectively, statistical methods are used to determine if the difference in means is significant or if it could just be due to chance.
  • The calculated mean difference was \(-0.45\), suggesting lower cholesterol levels on average for vegans.
  • The confidence interval for these means \((鈭�0.8773, 鈭�0.0227)\) did not include zero, indicating a statistically significant difference.

Such confidence intervals offer assurance in research, guiding whether observed differences may reflect true variances applicable to the broader public.

This comparison provides insight that could influence dietary recommendations for cholesterol management in specific populations like postmenopausal women.
Focus on Postmenopausal Women Study
The inclusion of postmenopausal women in this study is notably significant. This demographic often experiences changes in body chemistry that can impact cholesterol levels and overall health.
The transition into menopause can sometimes lead to increased cholesterol, putting them at higher risk for cardiovascular diseases.
Thus, understanding how different diets affect this group's cholesterol levels is crucial.

This study differentiates between vegan and omnivore diets, providing data that may suggest beneficial dietary adjustments.
Since the calculated confidence interval for the mean difference in cholesterol levels between them suggested a significant disparity鈥攚ith vegans typically exhibiting lower cholesterol levels鈥攖here's a potential recommendation for dietary adjustments.

Postmenopausal women could benefit from such findings, possibly opting for a diet that supports healthier cholesterol levels.
Such focused studies allow for more targeted public health recommendations, potentially aiding in the development of diet plans that address specific health concerns tied to menopause.

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