91Ó°ÊÓ

Use the data in the table below for sitting adult males and females (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. (Hint: Draw a graph in each case.) $$ \begin{aligned} &\text { Sitting Back-to-Knee Length (inches) }\\\ &\begin{array}{l|c|c|c} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \text { in. } & 1.1 \text { in. } & \text { Normal } \\ \hline \text { Females } & 22.7 \text { in. } & 1.0 \text { in. } & \text { Normal } \\ \hline \end{array} \end{aligned} $$ Instead of using \(0.05\) for identifying significant values, use the criteria that a value \(x\) is significantly high if \(P(x\) or greater \() \leq 0.025\) and a value is significantly low if \(P(x\) or less \() \leq 0.025 .\) Find the female back-to-knee length, separating significant values from those that are not significant. Using these criteria, is a female back-to-knee length of 20 in. significantly low?

Step by step solution

01

Understand the Problem

Identify the given data for females: the mean \(\mu = 22.7 \, \text{in.}\), the standard deviation \(\sigma = 1.0 \, \text{in.}\), and the normal distribution.

02

Define Criteria for Significance

A value x is significantly high if \(P(x)\) or greater \(\leq 0.025\) and significantly low if \(P(x)\) or less \(\leq 0.025\).

03

Calculate Standard Scores (Z-scores)

To determine if 20 in. is significantly low, calculate the Z-score using \(Z = \frac{x - \mu}{\sigma} = \frac{20 - 22.7}{1.0} = -2.7\).

04

Find Corresponding Probability

Using the Z-score table or a calculator, find the probability corresponding to \(Z = -2.7\). The probability is approximately \(P(Z < -2.7) = 0.0035\).

05

Compare with Criteria

Since \(P(Z < -2.7) = 0.0035\), which is less than 0.025, 20 in. is considered significantly low.

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Key Concepts

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

Understanding the Normal Distribution

The normal distribution is a key concept in statistics, especially in the field of anthropometry. It's a way of describing data that clusters around a central value with a symmetrical distribution on both sides. For instance, if we look at the back-to-knee length measurements for sitting adult males and females, we'll see that many measurements fall near their respective means (averages) and fewer as they move away. This pattern forms the classic 'bell curve' shape.
Key properties of a normal distribution are:

• Symmetry: The left side of the curve mirrors the right side.
• Mean, median, and mode are all the same value.
• 68% of the data falls within one standard deviation from the mean, 95% within two, and 99.7% within three.

Significance Levels Simplified

In statistics, 'significance levels' help us decide whether our results are meaningful. These levels show how confident we can be about our findings. For the given exercise, instead of using the common 0.05 significance level, we use a stricter criterion. A value is considered significantly high if the probability of it being that value or greater is 0.025 or less. Conversely, it's significantly low if the probability of it being that value or less is also 0.025 or less.
Think of significance levels as thresholds that help determine if an observed effect is rare under normal conditions. By using these criteria, we can make more informed decisions about our data.

Demystifying Z-Scores

Z-scores are a way to understand how far a particular value is from the mean, measured in terms of standard deviations. The formula for a Z-score is:
\( Z = \frac{x - \mu}{\sigma} \)
Where:

• \(x\) is the value we're interested in.
• \(\mu\) is the mean of the data.
• \(\sigma\) is the standard deviation.

In our example, we calculated the Z-score for a back-to-knee length of 20 inches to see if it's significantly low. Plugging in the values, we get:
\( Z = \frac{20 - 22.7}{1.0} = -2.7 \). This negative Z-score shows that 20 inches is 2.7 standard deviations below the mean.

Probability and Its Applications

Probability measures the likelihood of a specific event happening. We often use it in statistics to understand the chances of obtaining a certain result. With normal distributions, we can use Z-scores to find probabilities. For instance, to evaluate if a back-to-knee length of 20 inches is significantly low, we look at the probability of getting a Z-score of -2.7 or less using a Z-table or calculator. In this scenario, the probability is about 0.0035.
Since this probability is much lower than our 0.025 threshold, we conclude that 20 inches is indeed significantly low.
Understanding probability helps us make sense of data and decide if certain results are rare enough to be considered significant.