91Ó°ÊÓ

A study of the relationship between age and various visual functions (such as acuity and depth perception) reported the following observations on area of scleral lamina \(\left(\mathrm{mm}^{2}\right)\) from human optic nerve heads ("Morphometry of Nerve Fiber Bundle Pores in the Optic Nerve Head of the Human," Experimental Eye Research, 1988: 559–568): \(\begin{array}{lllllllll}2.75 & 2.62 & 2.74 & 3.85 & 2.34 & 2.74 & 3.93 & 4.21 & 3.88 \\ 4.33 & 3.46 & 4.52 & 2.43 & 3.65 & 2.78 & 3.56 & 3.01 & \end{array}\) a. Calculate \(\sum x_{i}\) and \(\sum x_{i}^{2}\). b. Use the values calculated in part (a) to compute the sample variance \(s^{2}\) and then the sample standard deviation \(s\).

Step by step solution

01

Calculate the Sum of Observations

To find \( \sum x_i \), add all the values together. There are 17 values in total.\[\sum x_i = 2.75 + 2.62 + 2.74 + 3.85 + 2.34 + 2.74 + 3.93 + 4.21 + 3.88 + 4.33 + 3.46 + 4.52 + 2.43 + 3.65 + 2.78 + 3.56 + 3.01 = 56.69\]

02

Calculate the Sum of Squared Observations

Now, calculate \( \sum x_i^2 \) by squaring each observation and then summing them up.\[\begin{align*}\sum x_i^2 &= 2.75^2 + 2.62^2 + 2.74^2 + 3.85^2 + 2.34^2 + 2.74^2 + 3.93^2 + 4.21^2 + 3.88^2 \& \quad + 4.33^2 + 3.46^2 + 4.52^2 + 2.43^2 + 3.65^2 + 2.78^2 + 3.56^2 + 3.01^2 \&= 3.40 + 3.86 + 7.52 + 14.82 + 5.48 + 7.52 + 15.44 + 17.73 + 15.05 \& \quad + 18.75 + 11.98 + 20.42 + 5.90 + 13.32 + 7.73 + 12.68 + 9.06 \&= 201.1434\end{align*}\]

03

Calculate the Mean of the Sample

The mean \( \overline{x} \) is calculated by dividing the sum of observations by the number of observations \( n \).\[\overline{x} = \frac{\sum x_i}{n} = \frac{56.69}{17} = 3.334\]

04

Compute the Sample Variance

Use the formula for sample variance:\[s^2 = \frac{1}{n-1} \left( \sum x_i^2 - \frac{(\sum x_i)^2}{n} \right)\]Substitute the calculated values:\[s^2 = \frac{1}{16} \left( 201.1434 - \frac{(56.69)^2}{17} \right)\]Calculate further:\[s^2 = \frac{1}{16} (201.1434 - 189.1791) = \frac{11.9643}{16} = 0.7478\]

05

Compute the Sample Standard Deviation

The sample standard deviation \( s \) is the square root of the sample variance.\[s = \sqrt{s^2} = \sqrt{0.7478} \approx 0.8647\]

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Sample Variance

The sample variance is a statistical measure that indicates how spread out the values in a data set are around the mean. It is particularly useful for understanding the extent of variability within a sample of data.
In our exercise, calculating the sample variance involves several clear steps:

• First, we compute the mean of the sample, which is the average value of all observations.
• Then, we employ the formula for sample variance: \[ s^2 = \frac{1}{n-1} \left( \sum x_i^2 - \frac{(\sum x_i)^2}{n} \right) \] where \( n \) is the number of observations.
• By substituting our calculated values into the formula, we arrive at the variance.

This variance tells us how much the individual data points deviate from the average. A larger variance indicates that the data points are more spread out. It’s an essential step in determining the standard deviation and provides insight into the diversity of the data set.

Sample Standard Deviation

The sample standard deviation is another key metric in statistical analysis. It is the square root of the sample variance and provides a measure of the amount of variation or dispersion in a set of values. The standard deviation is particularly helpful because it offers a direct interpretation of how much the data deviates from the mean.To compute it:

• Take the sample variance, which we've calculated previously.
• Find the square root of this variance: \[ s = \sqrt{s^2} \]
• This result shows us how closely each observation clusters around the mean on average.

The sample standard deviation is widely used in many fields because it provides a clear view of data consistency. Smaller values indicate that data points tend to be very close to the mean, while larger values suggest more variation.

Data Summation

Data summation is a fundamental concept in statistical analysis, representing the process of adding together all the data observations. This method is necessary for calculating both the mean and variance of a data set.
In our exercise, we perform two types of summations to facilitate further analysis:

• First, we calculate the sum of all data points, noted as \( \sum x_i \), by simply adding each value in the dataset.
• Second, we need the sum of squared data points, represented as \( \sum x_i^2 \). This involves squaring each data point and then adding them.

Both of these summation operations are crucial steps in the process of calculating variance and standard deviation. Understanding and executing data summation accurately enables effective statistical analysis, ensuring that subsequent calculations are based on precise data.