91Ó°ÊÓ

The data set mentioned in Exercise 1 also includes these third grade verbal IQ observations for males: \(\begin{array}{lllllllll}117 & 103 & 121 & 112 & 120 & 132 & 113 & 117 & 132 \\\ 149 & 125 & 131 & 136 & 107 & 108 & 113 & 136 & 114\end{array}\) and females \(\begin{array}{llllllll}114 & 102 & 113 & 131 & 124 & 117 & 120 & 90 \\ 114 & 109 & 102 & 114 & 127 & 127 & 103 & \end{array}\) Prior to obtaining data, denote the male values by \(X_{1}, \ldots, X_{m}\) and the female values by \(Y_{1}, \ldots, Y_{n}\) Suppose that the \(X_{i}\) 's constitute a random sample from a distribution with mean \(\mu_{1}\) and standard deviation \(\sigma_{1}\) and that the \(Y_{i}\) 's form a random sample (independent of the \(X_{i}\) 's) from another distribution with mean \(\mu_{2}\) and standard deviation \(\sigma_{2}\). a. Use rules of expected value to show that \(\bar{X}-\bar{Y}\) is an unbiased estimator of \(\mu_{1}-\mu_{2}\). Calculate the estimate for the given data. b. Use rules of variance from Chapter 6 to obtain an expression for the variance and standard deviation (standard error) of the estimator in part (a), and then compute the estimated standard error. c. Calculate a point estimate of the ratio \(\sigma_{1} / \sigma_{2}\) of the two standard deviations. d. Suppose one male third-grader and one female third-grader are randomly selected. Calculate a point estimate of the variance of the difference \(X-Y\) between male and female IQ.

Step by step solution

01

Determine the Expected Value

The estimator \( \bar{X} - \bar{Y} \) is unbiased if its expected value equals \( \mu_1 - \mu_2 \). Using the linearity of expectation, we have \( E(\bar{X}) = \mu_1 \) and \( E(\bar{Y}) = \mu_2 \). Thus, \( E(\bar{X} - \bar{Y}) = E(\bar{X}) - E(\bar{Y}) = \mu_1 - \mu_2 \). Therefore, \( \bar{X} - \bar{Y} \) is an unbiased estimator.

02

Calculate Sample Means

Compute the sample means for males and females. For males, \( \bar{X} = \frac{1}{18} \sum_{i=1}^{18} X_i = \frac{2336}{18} = 129.78 \). For females, \( \bar{Y} = \frac{1}{15} \sum_{i=1}^{15} Y_i = \frac{1713}{15} = 114.20 \). The point estimate of \( \mu_1 - \mu_2 \) is \( \bar{X} - \bar{Y} = 129.78 - 114.20 = 15.58 \).

03

Calculate the Variance of the Estimator

The variance of the difference \( \bar{X} - \bar{Y} \) is given by \( Var(\bar{X} - \bar{Y}) = \frac{\sigma_1^2}{m} + \frac{\sigma_2^2}{n} \). Since we don't know \( \sigma_1^2 \) and \( \sigma_2^2 \), we use sample variances as estimators. Calculate sample variance of males \( s_X^2 = \frac{1}{17} \sum_{i=1}^{18} (X_i - \bar{X})^2 = 141.8889 \) and females \( s_Y^2 = \frac{1}{14} \sum_{i=1}^{15} (Y_i - \bar{Y})^2 = 101.2714 \).

04

Calculate Standard Error

The standard error is \( \sqrt{\frac{s_X^2}{m} + \frac{s_Y^2}{n}} = \sqrt{\frac{141.8889}{18} + \frac{101.2714}{15}} = \sqrt{7.8827 + 6.7514} = \sqrt{14.634} \approx 3.826 \).

05

Estimate Ratio of Standard Deviations

To estimate the ratio \( \frac{\sigma_1}{\sigma_2} \), compute \( \frac{s_X}{s_Y} = \frac{ \sqrt{141.8889} }{ \sqrt{101.2714} } = \frac{11.9054}{10.0649} = 1.183 \).

06

Point Estimate of Variance of Difference

The variance of the difference \( X - Y \) when a single male and female are selected is \( s_X^2 + s_Y^2 = 141.8889 + 101.2714 = 243.1603 \).

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Key Concepts

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

Variance Calculation

Variance is a fundamental concept in statistics that measures how much the values in a data set deviate from the mean. It provides insight into the data's spread or consistency. To calculate the variance for a set of data points, you follow these steps:

• Find the mean (average) of the data set.
• Subtract the mean from each data point to get the deviation of each point.
• Square each deviation to remove negative values and emphasize larger differences.
• Sum all the squared deviations.
• Divide the total of squared deviations by the number of data points.

For sample variance, you'll divide by the number of observations minus one (n-1) due to Bessel's correction, which adjusts for bias in small sample sizes. This adjustment helps the sample variance to be an unbiased estimate of the population variance.
Therefore, understanding the calculation of variance is crucial for statistical estimation, as it plays a significant role in determining how data is distributed.

Unbiased Estimators

Unbiased estimators are statistical estimates that accurately reflect the true parameter of a population without systematic error. When an estimator is unbiased, its expected value equals the parameter it estimates. This means that on average, the estimator hits the true value of the parameter over many samples.
In the context of comparing two sample means—like male and female IQ scores—the estimator \( \bar{X} - \bar{Y} \) is unbiased if the expectation \( E(\bar{X} - \bar{Y}) = \mu_1 - \mu_2 \). This property is verified by using the linearity of expected value: \( E(\bar{X}) = \mu_1 \) and \( E(\bar{Y}) = \mu_2 \). Therefore, the expected value of the difference is \( E(\bar{X} - \bar{Y}) = \mu_1 - \mu_2 \).
Understanding and ensuring that your estimator is unbiased is vital because it ensures accurate reflection upon the population parameter, leading to credible and reliable conclusions from statistical data analysis.

Standard Error

Standard error quantifies the amount of variability or dispersion of a sample statistic, like the sample mean, over different samples. It helps in understanding the precision of the sample mean as an estimate of the population mean. The smaller the standard error, the more precise the estimate is.
For the difference between two means, the standard error can be calculated using the formula:

• \( SE = \sqrt{\frac{s_X^2}{m} + \frac{s_Y^2}{n}} \)

Where \( s_X^2 \) and \( s_Y^2 \) are the sample variances, and \( m \) and \( n \) are the sample sizes.
This formula arises because each sample's variance contributes independently to the variance of their difference. Thus, the standard error combines these variances to give a comprehensive measure of variability. It's crucial in constructing confidence intervals and hypothesis testing.

Sample Variance

Sample variance is a key indicator in statistics that measures how data points in a sample deviate from the sample mean. It serves as an estimate of the population variance when the full data set is unavailable.
To calculate the sample variance, use these steps:

• Calculate the mean of the sample (sample mean).
• Subtract the sample mean from each data point to find the deviation.
• Square each deviation to get rid of negative values.
• Sum these squared deviations.
• Divide by the number of observations minus one (n-1) to account for Bessel's correction.

This correction is crucial for making the sample variance an honest, unbiased estimation of the population variance. Recognizing and calculating sample variance accurately is important, as it underlies many statistical tests and calculations, providing the foundation for making inferences about population parameters.