Chapter 5 Numbers 511, 513, 515, And 518 Finding Proportions

Chapter 5 Numbers 511 513 515 And 518finding Proportions

Find the proportion of IQ scores on the Wechsler Adult Intelligence Scale (WAIS) that are above Kristen’s score of 125, below 82, within 9 points of the mean, and more than 40 points from the mean, given that scores approximate a normal distribution with a mean of 100 and a standard deviation of 15.

Determine the IQ scores corresponding to the upper 2 percent, the lower 10 percent, the upper 60 percent, the middle 95 percent, and the middle 99 percent of the WAIS scores, considering the same distribution parameters.

A researcher studying common cold sufferers estimates the duration of symptoms follows a normal distribution with a mean of 83 hours and a standard deviation of 20 hours. Find the estimated number of hours for the shortest 5 percent, the proportion of sufferers with symptoms longer than 48 hours, fewer than 61 hours, and for the extreme 1 percent above or below the mean. Also, determine the proportion of sufferers between 24 and 72 hours, the middle 95 percent duration, between 48 and 96 hours, and the hours suffering by the top 20 percent, the bottom 3 percent of durations, and by those suffering more than 61 hours. Calculate the proportion of sufferers exceeding 61 hours, and the proportion suffering exactly 61 hours.

In a study on BMI among American males, the distribution is positively skewed with a mean of 28 and a standard deviation of 4. Determine whether the median exceeds, equals, or is less than the mean; find the z-score that defines overweight (BMI between 25 and 29.9), and obese (BMI 30 or more).

A televised debate poll conducted after the 2012 presidential campaign between Barack Obama and Mitt Romney involved a telephone poll where viewers called in to register preferences. Comment on the randomness of this sample and how to improve it.

In a probability problem, the chance of a boy being born equals 0.50. For a family with two children, find the probabilities of: (a) two boys, (b) two girls, and (c) one boy and one girl.

Using a set of playing cards with symbols, calculate the probability of: (a) a correct guess on a single card, (b) two guesses in order, (c) three correct guesses in a row, and (d) three incorrect guesses.

A sensor monitors reactor performance with a 97% accuracy rate, but has a 2% false alarm probability and a 1% miss rate. Find the probability that the sensor gives an incorrect report, the probability of simultaneous false alarms when two independent sensors are used, and the probability of missed detection when either sensor reports excessive radiation.

In a breast cancer screening context, with a 1% prevalence rate, a mammogram's positive detection probability of 80%, a negative detection probability of 90%, compute the probability that a woman has cancer given a positive result, and the probability of not having cancer given a negative result, using Bayesian analysis.

Sample Paper For Above instruction

The analysis of proportions and scores within the context of normal distribution plays a vital role in psychological measurement, epidemiology, and health sciences. Particularly, IQ scores and health-related data often follow a bell-shaped curve that can be described and analyzed using the properties of the normal distribution and standard scores (z-scores). This paper explores various applications of normal distribution, including calculating proportions associated with certain scores, understanding percentile ranks, and applying Bayesian reasoning in medical testing scenarios.

Introduction

Understanding how to find proportions and scores within a normal distribution is fundamental in interpreting data from psychological assessments, health surveys, and diagnostic tests. The WAIS IQ scores are known to distribute approximately normally, with a mean of 100 and a standard deviation of 15 (Wechsler, 2008). Such an understanding allows psychologists and researchers to determine how an individual score compares to the population, assess the percentile rank, and interpret the significance of the results in context.

Finding Proportions of IQ Scores

Given the normal distribution of IQ scores with a mean (μ) of 100 and a standard deviation (σ) of 15, we can compute the proportion of scores above or below certain values using the standard normal table or Z-score calculations. For example, Kristen’s IQ score of 125 corresponds to a Z-score of (125 - 100) / 15 = 1.67. Using the standard normal distribution table, the proportion of scores above Kristen’s score is approximately 0.0475, or 4.75%. Similarly, an IQ score below 82, with a Z-score of (82 - 100) / 15 ≈ -1.20, corresponds to a cumulative proportion of about 0.1151, meaning 11.51% of scores fall below this value.

Within 9 points of the mean (from 91 to 109), the corresponding Z-scores are -0.60 and +0.60, respectively. The cumulative proportions at these Z-scores are approximately 0.2724 and 0.7257. The proportion within this range is then approximately 0.7257 - 0.2724 = 0.4533, or 45.33%. Scores more than 40 points from the mean, i.e., below 60 or above 140, correspond to Z-scores of -2.67 and +2.67, respectively, with cumulative proportions around 0.0038 and 0.9963; thus, about 0.38% of scores are more than 40 points from the mean.

Score Percentiles and Cutoffs

To find scores corresponding to specific percentiles, the inverse of the normal distribution is used. The upper 2 percent corresponds to a Z-score of approximately +2.05, which translates to IQ = 100 + 2.05(15) ≈ 130.75. The lower 10 percent corresponds to a Z-score of approximately -1.28, resulting in IQ ≈ 100 - 1.28(15) ≈ 80.8. The upper 60 percent aligns with locating the 40th percentile, with a Z-score of about -0.25, giving IQ ≈ 100 - 0.25(15) ≈ 96.25. The middle 95 percent spans from approximately the 2.5th to 97.5th percentiles, corresponding to Z-scores of about -1.96 and +1.96, which translates to IQ scores from approximately 70.6 to 129.4. Similarly, the middle 99 percent includes scores from about the 0.5th to 99.5th percentiles, with Z-scores of roughly -2.81 and +2.81, giving IQ scores from approximately 56.7 to 143.3.

Health Data Analysis Using Normal Distribution

In the case of cold duration estimates, where the mean is 83 hours and standard deviation 20 hours, the shortest 5 percent duration can be found by identifying the 5th percentile Z-score of approximately -1.64. Multiplying this Z-score by 20 and adding 83 yields 83 + (-1.64)(20) ≈ 50.8 hours. The proportion of sufferers experiencing more than 48 hours can be calculated by finding the Z-score for 48 hours: (48 - 83) / 20 = -1.75, which corresponds to about 0.0401, indicating approximately 96% suffer longer than 48 hours. Fewer than 61 hours corresponds to a Z-score of (61 - 83) / 20 = -1.1, with a cumulative proportion of about 0.1357, so about 86.43% suffer less than 61 hours.

The 1 percent extremes above and below the mean can be found with Z-scores of ±2.33; thus, hours corresponding to these Z-scores are approximately 83 + 2.33(20) = 127.6 hours and 83 - 2.33(20) = 38.4 hours. The proportion of durations between these two values approximates 98%, leaving about 2% in each extreme tail. Between 24 and 72 hours, the corresponding Z-scores are approximately -3.55 and -0.55, yielding proportions accordingly; these calculations help in understanding the distribution of cold durations for epidemiological insights.

Application in BMI and Public Health

BMI data with a mean of 28 and a standard deviation of 4 exhibit a positive skew, implying that the mean slightly exceeds the median, which is typical in such distributions. The Z-score for overweight BMI (25-29.9) has a lower boundary at (25 - 28) / 4 = -0.75, indicating that BMI scores below this Z-score are less common. Obesity, defined at BMI of 30 or more, corresponds to a Z-score of (30 - 28) / 4 = 0.50.

Evaluating Polls and Probabilities

Analyzing the validity of polls, such as a telephone survey post-debate, involves understanding sampling methods and potential biases—convenience sampling or self-selection could compromise randomness. Improving this involves implementing stratified random sampling or ensuring representativeness across demographic groups. Probability calculations for events like family children, card guesses, and sensor accuracy leverage basic rules of probability, including the multiplication rule for independent events and the complement rule for error rates. Bayesian reasoning in medical testing combines prior probabilities with test accuracies to determine the likelihood of disease given test results, which is essential in diagnostic decision-making.

Conclusion

The application of normal distribution principles across various fields—from intelligence testing to health diagnostics—provides powerful tools for quantitative analysis and decision making. Understanding how to compute proportions, define cutoffs, and interpret scores within the context of the normal curve enables professionals to make informed assessments about the population and individual cases alike. Furthermore, thoughtful application of probability concepts and Bayesian methods enhances the accuracy of medical diagnostics and the effectiveness of health interventions.

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