Question

The serum cholesterol levels of a population of 12 to 14-year-olds follow a normal distribution with mean \(155 \mathrm{mg} / \mathrm{dl}\) and standard deviation \(27 \mathrm{mg} / \mathrm{dl}\) (as in Example 4.1.1). (a) What percentage of the 12 - to 14 -year-olds have serum cholesterol values between 145 and \(165 \mathrm{mg} / \mathrm{dl} ?\) (b) Suppose we were to choose at random from the population a large number of groups of nine 12 - to 14-year-olds each. In what percentage of the groups would the group mean cholesterol value be between 145 and \(165 \mathrm{mg} / \mathrm{dl} ?\) (c) If \(\bar{Y}\) represents the mean cholesterol value of a random sample of nine 12 - to 14 -year-olds from the population, what is \(\operatorname{Pr}\\{145 \leq \bar{Y} \leq 165\\} ?\)

Short answer

Part (a): The percentage of 12 to 14-year-olds with serum cholesterol between 145 and 165 mg/dl can be found after calculating the corresponding Z-scores and using the standard normal distribution table. Part (b) and (c): The percentage of groups with a mean cholesterol between 145 and 165 mg/dl and the probability for a random sample are determined by calculating the standard error and Z-scores for the mean, then referring to the standard normal distribution table.

Step-by-step solution

Find Z-Scores for Part (a)

Convert the serum cholesterol values 145 mg/dl and 165 mg/dl to their corresponding Z-scores using the formula Z = (X - mean) / standard deviation, where X is the serum cholesterol value, mean is 155 mg/dl and standard deviation is 27 mg/dl.

Use Standard Normal Distribution Table for Part (a)

Look up the Z-scores obtained in Step 1 in the standard normal distribution table to find the probabilities that correspond to these Z-scores and calculate the percentage of individuals with cholesterol levels between 145 mg/dl and 165 mg/dl.

Find Standard Error for Part (b) and Part (c)

Calculate the standard error of the mean for a sample size of 9 using the formula Standard Error = standard deviation / sqrt(sample size).

Find Z-Scores for Part (b) and Part (c)

Convert the group mean cholesterol values 145 mg/dl and 165 mg/dl to their corresponding Z-scores using the newly calculated standard error.

Use Standard Normal Distribution Table for Part (b) and Part (c)

Look up the Z-scores obtained in Step 4 in the standard normal distribution table to find the probabilities that correspond to these Z-scores and calculate the percentage of groups with a mean cholesterol level between 145 mg/dl and 165 mg/dl.

Answer Part (c)

The probability that a random sample of nine 12 to 14-year-olds will have a mean cholesterol value between 145 mg/dl and 165 mg/dl is the same percentage calculated in Step 5.

Key concepts

Serum Cholesterol Levels

Understanding serum cholesterol levels is crucial when analyzing the health status of a population. In exercises like the one provided, we explore the distribution of these levels among a specific demographic—in this case, 12 to 14-year-olds. Cholesterol levels are measured in milligrams per deciliter (mg/dl), and a normal range is typically determined based on extensive health research. Monitoring these levels is essential as both high and low values can indicate potential health risks. Through statistical analysis, such as the one in our exercise, we gain insights on the percentage of a population that falls within a defined cholesterol range, helping to reveal the group's overall health pattern.

For instance, in our exercise, the average (mean) cholesterol level is given as 155 mg/dl, and we are interested in understanding what proportion of this age group falls within the healthy or expected range. This has real-world applications in public health and educational programs aimed at advocating for better lifestyle choices among adolescents.

Standard Deviation

Standard deviation is a statistical measure of the amount of variability or dispersion around the mean (average) of a set of data. In other words, it tells us how spread out the values in a dataset are. Having a low standard deviation indicates that the values tend to be close to the mean, whereas a high standard deviation signals that the values are spread out over a wide range. In the context of our exercise, a standard deviation of 27 mg/dl suggests that individual cholesterol levels in this population can vary quite significantly from the mean. This is an essential piece in calculating the Z-scores and understanding how much individual measurements deviate from the average, helping us assess the diversity of serum cholesterol levels within the group.

Z-Score

The Z-score is a statistical measure that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. It's calculated using the formula: \[ Z = \frac{(X - \text{mean})}{\text{standard deviation}} \]where X represents the value from the dataset. Z-scores allow us to standardize different sets of data, making them comparable. They also play a pivotal role in determining the probability of a value occurring within a standard normal distribution, which is particularly handy when dealing with variables like serum cholesterol levels, where it's necessary to establish how common or rare certain levels are within a given population.

Standard Normal Distribution Table

A standard normal distribution table, also known as a Z-table, is a reference table that provides the probability of a Z-score occurring within a standard normal distribution. This distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. By converting raw scores from our dataset into Z-scores, we can use the Z-table to determine the percentage of the population that falls between any two given cholesterol levels. This helps us answer questions like those in part (a) of the exercise, guiding students through the process of finding precise probabilities for standardized ranges of data.

Standard Error

The standard error is a statistical term that refers to the estimated standard deviation of a sampling distribution. It essentially measures the accuracy with which a sample represents a population. In practice, the standard error decreases as the sample size increases, implying that a larger sample gives a more accurate representation of the population mean. The formula to calculate the standard error is given by: \[ \text{Standard Error} = \frac{\text{standard deviation}}{\sqrt{\text{sample size}}} \]In our exercise, the standard error helps us determine the variability of the sample means around the population mean for groups of nine students. Consequently, we utilize it for converting cholesterol levels to Z-scores when the question shifts from individual measurements to group averages.

Sample Size

Sample size plays a crucial part in statistical analysis, impacting the reliability and precision of the results. It represents the number of observations or individuals from a population that are included in the sample. In the broader context, the greater the sample size, the smaller the standard error, resulting in more precise estimates of population parameters. In problems like part (b) of our exercise, the sample size affects the standard error and hence the Z-scores for the group means. When calculating probabilities for repeated sampling, as demonstrated in part (c), an adequate sample size is vital to ensure the validity of the conclusions drawn about the population's cholesterol levels.