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Chapter 2: Problem 24

Heights of 10 year olds. Heights of 10 year olds, regardless of gender, closely follow a normal distribution with mean 55 inches and standard deviation 6 inches. (a) What is the probability that a randomly chosen 10 year old is shorter than 48 inches? (b) What is the probability that a randomly chosen 10 year old is between 60 and 65 inches? (c) If the tallest \(10 \%\) of the class is considered "very tall", what is the height cutoff for "very tall"? (d) The height requirement for Batman the Ride at Six Flags Magic Mountain is 54 inches. What percent of 10 year olds cannot go on this ride?

Short Answer

Expert verified
(a) 12.1%, (b) 15.58%, (c) 62.68 inches, (d) 43.25%

Step by step solution

01

Understand Normal Distribution

Heights of 10-year-olds follow a normal distribution with a mean of 55 inches and a standard deviation of 6 inches. This is a bell-shaped curve symmetrical around the mean.
02

Convert to Z-score for Part (a)

To find the probability that a 10-year-old is shorter than 48 inches, we convert 48 inches to a Z-score. The Z-score formula is: \[ Z = \frac{X - \mu}{\sigma} \]where \( X \) is 48, \( \mu \) is the mean (55), and \( \sigma \) is the standard deviation (6). Substitute the values:\[ Z = \frac{48 - 55}{6} = -1.17 \]
03

Find Probability for Z-score in Part (a)

Using the Z-table, find the probability for \( Z = -1.17 \). This gives approximately \( P(Z < -1.17) = 0.121 \). Thus, the probability of a randomly chosen 10-year-old being shorter than 48 inches is 0.121.
04

Convert to Z-scores for Part (b)

Now find the probability that a 10-year-old is between 60 and 65 inches. Convert both to Z-scores:For 60 inches,\[ Z = \frac{60 - 55}{6} = 0.83 \]For 65 inches,\[ Z = \frac{65 - 55}{6} = 1.67 \]
05

Find Probability for Z-scores in Part (b)

Using the Z-table, find the probabilities:\[ P(Z < 0.83) = 0.7967 \]\[ P(Z < 1.67) = 0.9525 \]The probability between 60 and 65 inches is\[ P(0.83 < Z < 1.67) = P(Z < 1.67) - P(Z < 0.83) = 0.9525 - 0.7967 = 0.1558 \]
06

Determine Cutoff for Tallest 10% in Part (c)

The cutoff height for 'very tall' 10-year-olds is the 90th percentile height. From the Z-table, the Z-score for 90th percentile is approximately 1.28.Queue the Z-score formula for height:\[ Z = \frac{X - 55}{6} \to X = Z \times 6 + 55 = 1.28 \times 6 + 55 = 62.68 \]Thus, the cutoff height is approximately 62.68 inches.
07

Determine Percentage Unable for Ride in Part (d)

Heights less than 54 inches require a ride restriction.Calculate Z-score:\[ Z = \frac{54 - 55}{6} = -0.17 \]Use the Z-table to find\[ P(Z < -0.17) = 0.4325 \].Thus, approximately 43.25% of 10-year-olds cannot go on this ride.

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Key Concepts

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

Z-score
In the realm of statistics, the Z-score is a crucial concept that measures how many standard deviations an element is from the mean. It helps to transform data points to identify where they fall on a standard normal distribution. The formula for calculating the Z-score is given by:
  • \[ Z = \frac{X - \mu}{\sigma} \]
Where:
  • \( X \) is the data point you are evaluating.
  • \( \mu \) is the mean of the distribution.
  • \( \sigma \) is the standard deviation.
Imagine you want to know if a 10-year-old's height of 48 inches is typical or not. The Z-score tells us that it's \(-1.17\), which means the height is about 1.17 standard deviations below the average height of 55 inches. By referring to statistical tools like the standard normal distribution table, we can then find the probability associated with specific Z-scores.
Standard Deviation
Standard deviation, denoted by \( \sigma \), is a measure that expresses how much individual measurements or observations of a dataset deviate from the mean value. It quantifies the amount of variation or dispersion in a set of data points. A low standard deviation means data points are generally close to the mean, whereas a high standard deviation indicates the data points are spread out over a wider range.
  • If we take the example of the heights of 10-year-olds, a standard deviation of 6 inches tells us that individual heights usually hover around the mean, 55 inches, within this interval.
  • In our exercise, this helps to understand why some kids might be significantly taller or shorter than the most common heights.
Percentile
The concept of percentile ranks is a way of understanding how a particular measurement compares with other data in a distribution. The 90th percentile indicates that 90% of the data points are below it. This can be especially insightful in assessments or ranking situations.
In practical terms, if a 10-year-old's height reaches the 90th percentile, it means this child is taller than 90% of their peers. For our given normal distribution of heights, the 90th percentile corresponded to a Z-score of 1.28, resulting in a cutoff height of approximately 62.68 inches. Therefore, 10-year-olds who are 62.68 inches tall or above can be considered part of the tallest 10% of their group.
Probability
Probability is the measure of the likelihood that an event will occur. It ranges from 0 (impossible) to 1 (certain). Probabilities describe everyday situations in terms of how likely they are to happen.
  • A key aspect in our exercise is finding the probability of certain height ranges among 10-year-olds.
  • For instance, the chance of a child being shorter than 48 inches involves calculating the area under the normal distribution curve to the left of 48 inches.
  • This corresponds to a Z-score of \(-1.17\), which relates to a probability of approximately 0.121 or 12.1%.
  • Similarly, knowing that 43.25% of 10-year-olds are too short for a ride indicates the probability of them being under the stipulated height limit.

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Most popular questions from this chapter

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