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A previous study found that people consuming large quantities of vegetables containing lutein (mainly spinach) were less likely to develop macular degeneration, a common eye disease among older people (age \(65+)\) that causes a substantial loss in visual acuity and in some cases can lead to total blindness. To follow up on this observation, a clinical trial is planned in which participants 65+ years of age without macular degeneration will be assigned to either a high-dose lutein supplement tablet or a placebo tablet taken once per day. To estimate the possible therapeutic effect, a pilot study was conducted in which 9 people \(65+\) years of age were randomized to placebo and 9 people \(65+\) years of age were randomized to lutein tablets (active treatment). Their serum lutein level was measured at baseline and again after 4 months of follow-up. From previous studies, people with serum lutein \(\geq 10 \mathrm{mg} / \mathrm{dL}\) are expected to get some protection from macular degeneration. However, the level of serum lutein may vary depending on genetic factors, dietary factors, and study supplements. Suppose that among people randomized to lutein tablets, at a 4-month follow-up the mean serum lutein level \(=21 \mathrm{mg} / \mathrm{dL}\) with standard deviation \(=8 \mathrm{mg} / \mathrm{dL} .\) If we presume a normal distribution for serum-lutein values among lutein-treated participants, then what percentage of people randomized to lutein tablets will have serum lutein in the therapeutic range?

Step by step solution

01

Calculate the Z-score

The Z-score is a measure of how many standard deviations an element is from the mean. It is calculated using the formula: \[ Z = \frac{X - \mu}{\sigma} \] where \( X = 10 \) mg/dL (the lower threshold for protection), \( \mu = 21 \) mg/dL (the mean serum lutein level), and \( \sigma = 8 \) mg/dL (the standard deviation). Plugging in the values, we get: \[ Z = \frac{10 - 21}{8} = \frac{-11}{8} = -1.375 \] This tells us how far 10 mg/dL is from the mean in terms of standard deviations.

02

Use the Z-score to find the percentile

Using a standard normal distribution table, we can determine the percentage of lutein-treated individuals who fall below 10 mg/dL. For a Z-score of -1.375, the table shows approximately 0.0843 or 8.43%. This means 8.43% of individuals have a serum lutein level below 10 mg/dL.

03

Calculate the therapeutic range percentage

Since 8.43% of individuals have serum lutein levels below the therapeutic range, the remaining percentage will be those in the therapeutic range or above. Therefore, the percentage of people in the therapeutic range is:\[ 100\% - 8.43\% = 91.57\% \].

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Key Concepts

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

Clinical Trial

A clinical trial is a research study conducted to evaluate the effects of medical, surgical, or behavioral interventions in humans. This type of study helps to determine the efficacy and safety of new treatments or existing treatments that require further investigation.

Clinical trials are often conducted in multiple phases, each designed to answer specific questions:

• Phase I: Assesses safety and dosage.
• Phase II: Tests efficacy and side effects.
• Phase III: Confirms effectiveness, monitors side effects, and compares with standard treatments.
• Phase IV: Involves post-market surveillance to assess long-term effects.

In the context of the macular degeneration study, participants are randomly assigned to receive either a high-dose lutein or a placebo. Randomization helps eliminate bias, ensuring a fair comparison between the treatment and control groups.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetrical and described by its mean and standard deviation. This bell-shaped curve is crucial in the field of statistics, particularly because of its properties:

• Most of the data lies around the mean (or average).
• Approximately 68% falls within one standard deviation of the mean.
• 95% is within two standard deviations.
• 99.7% falls within three standard deviations.

Assumptions of normal distribution are often applied in biostatistics and clinical trials, as seen in the exercise on lutein levels among participants. Here, the distribution of serum lutein levels allows researchers to apply normal distribution principles to determine the proportion of individuals in a specific range.

Z-score

A Z-score is a statistical measurement that tells you how many standard deviations a data point is from the mean of the data set. This is a valuable tool in biostatistics for assessing the relative position of a single data point within a distribution.

To calculate a Z-score, use the formula: \[ Z = \frac{X - \mu}{\sigma} \]where:

• X is the value being evaluated (in our case, 10 mg/dL for serum lutein).
• \(\mu\) is the mean of the data set (21 mg/dL).
• \(\sigma\) is the standard deviation of the data set (8 mg/dL).

A Z-score helps in comparing scores from different normal distributions, and in this study, it was used to determine how far a serum lutein level of 10 mg/dL was from the mean.

Macular Degeneration

Macular degeneration is an eye disease that affects the macula, the central part of the retina, leading to loss of vision. It primarily affects older adults and comes in two main forms:

• Dry macular degeneration: More common and less severe, it involves the thinning of the macula and the accumulation of drusen.
• Wet macular degeneration: Less common but more severe, it involves the growth of abnormal blood vessels under the retina.

Lifestyle and nutritional factors can influence disease progression. As the initial study suggested, higher serum lutein levels, commonly found in leafy greens like spinach, may offer protective benefits against this condition.

Statistical Analysis

Statistical analysis is the process of collecting and analyzing data to identify patterns, relationships, or trends. In clinical research, it's essential for interpreting complex data sets arising from trials and studies.

This involves a range of methods, from descriptive statistics to inferential statistics, allowing researchers to:

• Summarize data sets with measures like mean, median, and mode.
• Perform hypothesis testing to make inferences about a population.
• Assess relationships between variables using techniques like regression analysis.

In the context of our exercise, statistical analysis involved calculating the Z-score and using the normal distribution table to interpret the percentage of participants with therapeutic serum lutein levels.