Question

A certain assay for serum alanine aminotransferase (ALT) is rather imprecise. The results of repeated assays of a single specimen follow a normal distribution with mean equal to the true ALT concentration for that specimen and standard deviation equal to \(4 \mathrm{U} / \mathrm{l}\) (see Example 2.2 .12 ). Suppose that a certain hospital lab measures many specimens every day, performing one assay for each specimen, and that specimens with ALT readings of \(40 \mathrm{U} / \mathrm{l}\) or more are flagged as "unusually high." If a patient's true ALT concentration is \(35 \mathrm{U} / \mathrm{l}\), what is the probability that his specimen will be flagged as "unusually high"?

Short answer

The probability that a patient's specimen with a true ALT concentration of 35 U/l will be flagged as 'unusually high' is approximately 0.1056 or 10.56%.

Step-by-step solution

- Determine the Z-score

To find the probability that the ALT reading will be 'unusually high,' first calculate the Z-score. The Z-score is the number of standard deviations a data point is from the mean. The formula for the Z-score is: \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value of interest (40 U/l in this case), \( \mu \) is the mean (35 U/l), and \( \sigma \) is the standard deviation (4 U/l).

- Calculate the Z-score

Using the Z-score formula from Step 1: \( Z = \frac{40 - 35}{4} = \frac{5}{4} = 1.25 \). This means that an ALT reading of 40 U/l is 1.25 standard deviations above the mean.

- Find the probability from the Z-table

Using a standard normal distribution table or a calculator, find the probability that a standard normal variable is greater than or equal to the Z-score of 1.25. This will give the probability of a patient's ALT concentration being flagged as 'unusually high.'

- Interpret the Z-table result

The result from the Z-table gives the probability of a value being less than the Z-score. To find the probability of being greater than or equal to the Z-score, subtract the table value from 1. For a Z-score of 1.25, the table value is approximately 0.8944. Therefore, the probability is \(1 - 0.8944 = 0.1056\).

Key concepts

Z-score Calculation

The Z-score is a crucial concept in statistics, allowing us to understand the position of a data point within a normal distribution. It represents the number of standard deviations a certain value, such as a test result, is away from the mean average value.

Let's break it down with an example. Suppose you've taken a test and received a score that's above average. To know exactly how well you did, you calculate the Z-score. This score will tell you whether you're just a little above average or way off the charts! In the case of our ALT assay, the Z-score formula \( Z = \frac{X - \mu}{\sigma} \) becomes super handy. It translates the ALT level of a patient from U/l into a standardized score that tells us how far from the typical result it is. When we calculated \( Z = \frac{40 - 35}{4} = 1.25 \) for our patient, we discovered their result is 1.25 standard deviations above the mean—the score is out of the ordinary, but just how uncommon is it? That's where probability and understanding of normal distribution come into play.

Probability

Probability is all about measuring the chance of something happening. It ranges from 0 (no chance) to 1 (certainty). In our daily life, we estimate probabilities all the time. Think about it - predicting the chance of rain today, or whether you'll win a game. In our ALT assay example, we calculated a probability to figure out how likely it is that a patient's test result will be flagged as 'unusually high.'

To grasp this, imagine you're betting on which marble you'll pick from a bag of mixed colors. Finding the probability of getting a red one involves counting how many red marbles are there compared to the total number. In the medical test scenario, after using the Z-score, we looked for the probability corresponding to that score to answer our question. Using statistical tables or tools, we found the likelihood a patient's ALT level will be rare enough to raise a red flag. Specifically, we got a probability of 0.1056, meaning there's about a 10.56% chance the patient's result is 'unusually high.' It's like saying, out of 100 patients with the same true ALT concentration, about 11 might get a result that stands out enough to be noticed.

Normal Distribution

Normal distribution, often pictured as a bell curve, is a way to represent how data is dispersed. Lots of things in nature and human activity fall into this pattern, where most values are around a middle point and fewer and fewer are found the farther away you get from this point.

Think of it like this: when kids line up by height, there's a bunch of them right around the middle height, and only a few really tall or really short ones. That's the essence of normal distribution. In our ALT example, the normal distribution describes the spread of repeated test results—a few will be low, most will be around the true ALT level (the mean), and a few will be high.

Connecting to Our Example

Once we've calculated the Z-score for a patient’s ALT level, we can use the properties of the normal distribution to determine how unusual this score is. Since normal distribution is symmetric, values farther from the mean happen less often, which is why an ALT level of 40 U/l is not a common result for a person with a true level of 35 U/l. The symmetry also helps when we use the Z-score to determine the probability of this 'unusually high' reading, relying on the standardized distribution to give us the answers we need.