91Ó°ÊÓ

Assume that college women's heights are approximately Normally distributed with a mean of 65 inches and a standard deviation of \(2.5\) inches. On the horizontal axis of the graph, indicate the heights that correspond to the \(z\) -scores provided. (See the labeling in Exercise 6.14.) Use only the Empirical Rule to choose your answers. Sixty inches is 5 feet, and 72 inches is 6 feet. a. Roughly what percentage of women's heights are greater than \(72.5\) inches? i. almost all iii. \(50 \%\) v. about \(0 \%\) ii. \(75 \%\) iv. \(25 \%\) b. Roughly what percentage of women's heights are between 60 and 70 inches? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) c. Roughly what percentage of women's heights are between 65 and \(67.5\) inches? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) d. Roughly what percentage of women's heights are between \(62.5\) and \(67.5\) inches? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) e. Roughly what percentage of women's heights are less than \(57.5\) inches? i. almost all iii. \(68 \%\) v. about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) f. Roughly what percentage of women's heights are between 65 and 70 inches? i. almost all iii. \(47.5 \%\) v. \(2.5 \%\) ii. \(95 \%\) iv. \(34 \%\)

Step by step solution

01

Identify Mean and Standard Deviation

Here the mean is given as 65 inches and the standard deviation is 2.5 inches.

02

Calculate Z-Scores and Use the Empirical Rule for Part a

To solve for part a, first calculate the Z-score for 72.5 inches. The Z-score formula is \(Z = (X - \mu) /\sigma\), where \(X\) is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Substituting the given values, Z score = \((72.5 - 65) / 2.5\) gives Z = 3. This means that 72.5 inches is 3 standard deviations away from the mean. Therefore, a very small percentage of women's heights will be greater than 72.5 inches. The closest match from the given choices is 'about 0%'.

03

Use the Empirical Rule for Part b

To solve part b, check between what standard deviations 60 and 70 inches fall. 60 inches and 70 inches are 2 standard deviations away from the mean. Based on the empirical rule, about 95% of the data falls within 2 standard deviations of the mean. Therefore, roughly 95% of women's heights are between 60 and 70 inches.

04

Use the Empirical Rule for Part c

For part c, calculate the standard deviations 65 and 67.5 inches fall. Both these heights are within one standard deviation from the mean and according to empirical rule, 68% of women's heights are between these two values.

05

Use the Empirical Rule for Part d

For part d, note that both 62.5 and 67.5 inches falls within one standard deviation of the mean. Therefore, about 68% of data falls within one standard deviation of the mean.

06

Calculate Z-Scores and Use the Empirical Rule for Part e

For part e, calculate the Z-score 57.5 inches, which gives us Z = -3. This means that 57.5 inches is 3 standard deviations away from the mean. Therefore, a very small percentage of women's heights will be less than 57.5 inches. Our closest match from the given choices is 'about 0%'.

07

Use the Empirical Rule for Part f

For part f, consider the range 65 and 70 inches, which lie within 2 standard deviations from the mean. However, since we're only considering the range above the mean, it would only be half of the total percentage that fall within 2 standard deviations from the mean. Therefore, approximately 47.5% of women's heights are between 65 and 70 inches.

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Key Concepts

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

Empirical Rule

The Empirical Rule is an important concept in statistical analysis and helps us understand how data is distributed within a normal distribution. It states that:

• About 68% of the data falls within 1 standard deviation of the mean.
• Approximately 95% falls within 2 standard deviations.
• Nearly 99.7% lies within 3 standard deviations.

This rule is extremely useful because it allows us to quickly estimate the probability of a data point falling within a particular range of values, just by knowing the mean and standard deviation of a normal distribution. It's often depicted as a bell curve, where more data points are closer to the mean and fewer as we move further away.

Z-score Calculation

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It shows how many standard deviations a particular data point is from the mean: \[ Z = \frac{X - \mu}{\sigma} \] Where:

• \( X \) is the value in question.
• \( \mu \) is the mean of the data set.
• \( \sigma \) is the standard deviation.

Calculating a Z-score allows you to determine the position of a value within a distribution. If the Z-score is 0, it indicates the value is exactly average. A Z-score of 1 means it's one standard deviation above average, while a Z-score of -1 shows it’s one standard deviation below the mean. This is particularly useful for comparing data points from different distributions.

Standard Deviation

The standard deviation is a key measure of variability or dispersion in a data set. It tells us how much the individual data points deviate from the mean. The formula for calculating standard deviation is: \[ \sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}} \] Where:

• \( X \) represents each value in the data set.
• \( \mu \) is the mean of the data set.
• \( N \) is the number of data points.

A small standard deviation indicates that the data points are close to the mean, while a large standard deviation signifies that the data points are spread out over a wider range. Understanding standard deviation helps assess the risk or volatility in various contexts, such as finance, manufacturing, and other fields of study.

Statistical Analysis

Statistical analysis involves collecting, examining, and interpreting data to discover patterns or trends. It plays a critical role in decision-making processes across various fields. Here are some key steps in statistical analysis:

• **Data Collection:** Gather data through surveys, experiments, or observational studies.
• **Descriptive Statistics:** Summarize the data using measures like mean, median, mode, and standard deviation.
• **Inferential Statistics:** Make inferences or predictions about a population based on a sample of data, often involving hypothesis testing.

Statistical analysis can help to validate models, forecast future trends, or determine relationships between variables. It relies heavily on concepts such as normal distribution, Z-scores, and standard deviation to ensure accurate and reliable results.