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Chapter 10: Problem 49

A study was performed among 40 boys in a school in Edinburgh to look at the presence of spermatozoa in urine samples according to age \([15] .\) The boys entered the study at \(8-11\) years of age and left the study at \(12-18\) years of age. A 24-hour urine sample was supplied every 3 months by each boy. Table 10.28 gives the presence or absence of sperm cells in the urine samples for each boy together with the ages at entrance and exit of the study and the age at the first sperm-positive urine sample. For all parts of this question, exclude boys who exited this study without 1 sperm-positive urine sample (i.e., boys 8,9,14,25,28,29,30) Suppose mean age at spermatogenesis = 13.67 years, with standard deviation \(=0.89\) years and we assume that the age at spermatogenesis follows a normal distribution. The pediatrician would like to know what is the earliest age (in months) before which \(95 \%\) of boys experience spermatogenesis because he or she would like to refer boys who haven't experienced spermatogenesis by this age to a specialist for further follow-up. Can you estimate this age from the information provided in this part of the problem?

Short Answer

Expert verified
The earliest age before which 95% of boys experience spermatogenesis is 173 months.

Step by step solution

01

Understand the Normal Distribution

The problem states that the age at spermatogenesis follows a normal distribution with a mean of 13.67 years and a standard deviation of 0.89 years. We need to find the age where 95% of the boys have experienced spermatogenesis. This equates to identifying a specific percentile (the 95th percentile) of the normal distribution.
02

Identify the Z-Score for the 95th Percentile

For a normal distribution, the 95th percentile corresponds to a Z-score of approximately 1.645 (from Z-tables). This Z-score indicates how many standard deviations away from the mean the 95th percentile is.
03

Calculate the Corresponding Age

Use the formula for converting a Z-score to a data value: \[ X = ext{mean} + Z imes ext{standard deviation} \]Substitute the values to find the age:\[ X = 13.67 + 1.645 \times 0.89 \]Calculate the result.
04

Convert the Age from Years to Months

First, compute the age in years from the expression in Step 3. Then, convert the age from years to months by multiplying the age by 12, since there are 12 months in a year. This will give us the age in months.

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Key Concepts

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

Normal Distribution
In the study of age at spermatogenesis, a normal distribution is used to model the ages at which boys first show signs of sperm production. The normal distribution is a fundamental concept in statistics characterized by its symmetrical, bell-shaped curve. It is determined by two critical parameters: the mean and the standard deviation. The mean is the arithmetic average of all ages, which in our study is 13.67 years. It represents the center of our distribution. The standard deviation, here being 0.89 years, gives us an idea of the spread of our data points around the mean. One of the key properties of a normal distribution is that it allows us to make probabilistic statements about the data. For this study, we are interested in the 95th percentile, meaning the age before which 95% of the boys are expected to have experienced spermatogenesis. The 95th percentile is found a specific number of standard deviations- typically represented by a Z-score - above the mean.
Pediatric Study
A pediatric study typically deals with health data concerning children and adolescents. In this study, researchers collected data on the presence of spermatozoa among boys aged between 8-18 years. Every three months, urine samples were collected to monitor when each boy first showed signs of spermatogenesis. Pediatric studies like these are crucial for understanding developmental milestones and guiding health interventions. The goal of this study was to determine an appropriate reference age by which boys should meet this developmental milestone. Those who do not may require further health evaluations as a precautionary measure. The study's ethical considerations include ensuring the privacy and confidentiality of the participants' data, particularly given the sensitivity of adolescent health records.
Statistical Analysis
To determine the age at which 95% of the boys in this study first experience spermatogenesis, a statistical analysis was conducted using the data collected. This involves calculating percentiles in the context of a normal distribution. A key step is finding the Z-score that corresponds to the 95th percentile.The Z-score for this percentile is approximately 1.645. Once the Z-score is identified, it is used in conjunction with the mean and standard deviation to calculate the corresponding age. The formula to convert a Z-score to an age is: \[ X = \text{mean} + Z \times \text{standard deviation} \]Substituting the given mean and standard deviation yields the age of interest. Such calculations allow researchers to make informed decisions about which participants might need additional follow-up.
Age at Spermatogenesis
Spermatogenesis refers to the process by which male germ cells develop into mature spermatozoa. The age at which boys undergo spermatogenesis can vary but often signals the onset of puberty. In this study, researchers aimed to identify the age by which a majority of boys reach this milestone. The study reveals a mean age of 13.67 years for spermatogenesis, with the calculation for the 95th percentile, or the earliest age before which 95% have experienced it, being critical for clinical decision-making. By using a normal distribution model, it was possible to determine this specific age in months, providing useful information for pediatricians to decide when a referral to a specialist might be warranted. Understanding this age range helps healthcare professionals monitor adolescent development and detect any anomalies early on.

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Most popular questions from this chapter

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