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Chapter 7: Problem 45

In the population of hospitals, the correlation of the number of beds and the number of discharges is \(\rho=.91 \text { (Example D of Section } 7.4) .\) To see how \(\operatorname{Var}\left(\bar{Y}_{R}\right)\) would be different if the correlation were different, plot \(\operatorname{Var}\left(\bar{Y}_{R}\right)\) for \(n=64\) as a function of \(\rho\) for \(-1<\rho<1\)

Short Answer

Expert verified
\(\operatorname{Var}(\bar{Y}_R)\) decreases as \(|\rho|\) approaches 1, and is maximal when \(\rho = 0\).

Step by step solution

01

Understand the Problem

We need to find how the variance of the average of a sample of size 64, \( \operatorname{Var}(\bar{Y}_R) \), changes with different correlation values \( \rho \) between the number of beds and the number of discharges in hospitals.
02

Recall Formula for Variance of Regression Variable

The variance of the average response in a regression with a correlation \( \rho \) can be given by \[\operatorname{Var}(\bar{Y}_R) = \sigma^2 \left( \frac{1}{n} - \frac{\rho^2}{s_x^2} \right)\]where \(n\) is the sample size, and \(\sigma^2\) and \(s_x^2\) are population variance and variance of predictors respectively. This can change given additional information.
03

Derive the Relevant Expression

In this scenario, based on the provided context and common linear regression frameworks, the formula simplifies to accommodate for \(n=64\) and changes in \(\rho\), giving the expression:\[\operatorname{Var}(\bar{Y}_R) = \frac{\sigma^2}{64} \cdot (1 - \rho^2)\]Here, the variance of the mean depends linearly on \(1 - \rho^2\).
04

Plot \( \operatorname{Var}(\bar{Y}_R) \)

To graph this for \(-1 < \rho < 1\):1. Create a range of \( \rho \) values from just above -1 to just below 1.2. Calculate \( \operatorname{Var}(\bar{Y}_R) \) for each value using the formula \( \operatorname{Var}(\bar{Y}_R) = \frac{\sigma^2}{64} \cdot (1 - \rho^2) \).3. Plot these points, typically with \( \rho \) on the x-axis and \(\operatorname{Var}(\bar{Y}_R)\) on the y-axis.
05

Analyze the Plot

On the plot, \(\operatorname{Var}(\bar{Y}_R)\) decreases as \(\rho\) nears -1 or 1 because \((1 - \rho^2)\) approaches 0. Maximum variance is at \(\rho = 0\).

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Key Concepts

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

Variance
Variance measures how much a set of numbers differ from their average value. It tells us how spread out the numbers, like hospital discharges, are from the average.
For regression analysis, understanding variance helps us determine the reliability of our predictions. In general, high variance means that the numbers vary a lot from the average, indicating more uncertainty in predictions.
  • Mathematically, variance \(\sigma^2\) is the average of the squared differences from the mean.
  • In the exercise, we are specifically looking at the variance of the sample average, denoted as \operatorname{Var}(\bar{Y}_R)\, based on a sample size of 64.

The formula provided simplifies to \operatorname{Var}(\bar{Y}_R) = \frac{\sigma^2}{n}(1 - \rho^2)\. Here, the key takeaway is that variance in this context depends both on the sample size and the correlation factor \(\rho\). As variance gets smaller, it implies more precise predictions.
Regression
Regression is a statistical method that allows us to examine the relationship between two or more variables. In simple terms, it's about how one thing might affect another. For example, how does the number of beds in a hospital affect the number of discharges?
This task is typically accomplished through a mathematical equation that describes the relationship.
  • It's critical in many fields, such as economics, biology, and social sciences, as it helps in making predictions.
  • In the context of the given exercise, regression is used to predict the average number of discharges from the number of beds, based on historical data.

Understanding regression means understanding dependencies between variables, which can be exploited to make informed decisions based on predictions derived from regression models.
Sample Size 64
Sample size refers to the number of observations included in a sample. In the context of the exercise, the sample size is 64.
This means we are taking 64 different observations (like 64 hospitals) to make our calculations and predictions. Why is sample size important?
  • Larger sample sizes tend to provide more accurate estimates of the population parameters.
  • A sample size of 64 is substantial enough to gain a reasonable level of confidence in the statistical inferences drawn from the sample.

By using a sample size of 64, we're looking to minimize errors in our estimations regarding the correlation and variance, thereby making more robust conclusions.
Linear Regression
Linear regression is a way to model the relationship between two variables by fitting a linear equation to the observed data. Simply put, it's a line of best fit through data points with one variable as the dependent variable and the other as independent.
Key points:
  • The simplest form of regression, called *simple linear regression*, involves just two variables.
  • We calculate the relationship of a dependent variable (like discharges) with an independent variable (like the number of beds).

In our example, the linear regression equation might look like this: \(Y = a + bX\), where \(Y\) is the number of discharges, \(X\) is the number of beds, and \(a\), \(b\) are constants to be determined. Importantly, linear regression can provide insights on both the strength and direction of relationships between variables.

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Most popular questions from this chapter

Two surveys were independently conducted to estimate a population mean, \(\mu\) Denote the estimates and their standard errors by \(\bar{X}_{1}\) and \(\bar{X}_{2}\) and \(\sigma_{\bar{X}_{1}}\) and \(\sigma_{\bar{X}_{2}}\) Assume that \(\bar{X}_{1}\) and \(\bar{X}_{2}\) are unbiased. For some \(\alpha\) and \(\beta,\) the two estimates can be combined to give a better estimator: $$X=\alpha \bar{X}_{1}+\beta \bar{X}_{2}$$ a. Find the conditions on \(\alpha\) and \(\beta\) that make the combined estimate unbiased. b. What choice of \(\alpha\) and \(\beta\) minimizes the variances, subject to the condition of unbiasedness?

True or false? a. The center of a \(95 \%\) confidence interval for the population mean is a random variable. b. A \(95 \%\) confidence interval for \(\mu\) contains the sample mean with probability .95 c. A \(95 \%\) confidence interval contains \(95 \%\) of the population. d. Out of one hundred \(95 \%\) confidence intervals for \(\mu, 95\) will contain \(\mu\).

A simple random sample of a population of size 2000 yields the following 25 values: \(\begin{array}{rrrrr}104 & 109 & 111 & 109 & 87 \\ 86 & 80 & 119 & 88 & 122 \\\ 91 & 103 & 99 & 108 & 96 \\ 104 & 98 & 98 & 83 & 107 \\ 79 & 87 & 94 & 92 & 97\end{array}\) a. Calculate an unbiased estimate of the population mean. b. Calculate unbiased estimates of the population variance and \(\operatorname{Var}(\bar{X})\) c. Give approximate \(95 \%\) confidence intervals for the population mean and total.

A photograph of a large crowd on a beach is taken from a helicopter. The photo is of such high resolution that when sections are magnified, individual people can be identified, but to count the entire crowd in this way would be very time consuming. Devise a plan to estimate the number of people on the beach by using a sampling procedure.

Two populations are surveyed with simple random samples. A sample of size \(n_{1}\) is used for population I, which has a population standard deviation \(\sigma_{1} ;\) a sample of size \(n_{2}=2 n_{1}\) is used for population II, which has a population standard deviation \(\sigma_{2}=2 \sigma_{1} .\) Ignoring finite population corrections, in which of the two samples would you expect the estimate of the population mean to be more accurate?
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