91Ó°ÊÓ

In an article in IEEE Transactions on Instrumentation and Measurement \((2001,\) Vol. \(50,\) pp. \(986-990),\) researchers studied the effects of reducing current draw in a magnetic core by electronic means. They measured the current in a magnetic winding with and without the electronics in a paired experiment. Data for the case without electronics are provided in the following table. $$\begin{array}{cc}\hline & \text { Current Without } \\\\\text { Supply Voltage } & \text { Electronics (mA) } \\\\\hline 0.66 & 7.32 \\\1.32 & 12.22 \\\1.98 & 16.34 \\\2.64 & 23.66 \\\3.3 & 28.06 \\\3.96 & 33.39 \\\4.62 & 34.12 \\\3.28 & 39.21 \\\5.94 & 44.21 \\\6.6 & 47.48 \\\\\hline\end{array}$$ (a) Graph the data and fit a regression line to predict current without electronics to supply voltage. Is there a significant regression at \(\alpha=0.05 ?\) What is the \(P\) -value? (b) Estimate the correlation coefficient. (c) Test the hypothesis that \(\rho=0\) against the alternative \(\rho \neq 0\) with \(\alpha=0.05 .\) What is the \(P\) -value? (d) Compute a \(95 \%\) confidence interval for the correlation coefficient.

Step by step solution

01

Plot the Data

Begin by plotting the supply voltage on the x-axis and the current without electronics on the y-axis. This scatter plot will help visualize the linear relationship between the two variables.

02

Fit a Regression Line

Using the least squares method, fit a linear regression line of the form \( y = mx + c \) where \( y \) is the current without electronics, \( x \) is the supply voltage, \( m \) is the slope, and \( c \) is the y-intercept.

03

Compute Regression Statistics

Calculate the regression statistics, including the slope, intercept, and the coefficient of determination \( R^2 \). This will help in determining the strength and nature of the relationship.

04

Perform Significance Test

Conduct a hypothesis test to determine if the regression is significant. Here, the null hypothesis \( H_0 : \beta = 0 \) (no relationship) is tested against the alternative \( H_1 : \beta eq 0 \) using a t-test on the slope. If the p-value < 0.05, reject \( H_0 \). Record the p-value.

05

Calculate the Correlation Coefficient

Compute the Pearson correlation coefficient, \( r \), which measures the strength and direction of the linear relationship between the supply voltage and current without electronics.

06

Hypothesis Test for Correlation

Test the hypothesis \( H_0: \rho = 0 \) against \( H_1: \rho eq 0 \) using the correlation coefficient. Use a t-distribution to determine if \( r \) is significantly different from zero at \( \alpha = 0.05 \). Calculate and record the p-value.

07

Confidence Interval for Correlation

Calculate a 95% confidence interval for the correlation coefficient \( r \). Use Fisher's Z transformation to find the interval, then convert back to the correlation scale. This provides an interval estimate of the true correlation.

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Key Concepts

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

Hypothesis Testing

In hypothesis testing, we explore the validity of a claim using statistical methods. The null hypothesis (

• \( H_0 \)

) represents a statement with no effect or no difference. For linear regression, the hypothesis often examined is \( H_0: \beta = 0 \), which means there is no relationship between the variables.
The alternative hypothesis (

• \( H_1 \)

) suggests a relationship exists, so \( H_1: \beta eq 0 \).To test this, we use a t-test which involves calculating the p-value. If the p-value is less than the significance level, \( \alpha \) (commonly 0.05), we reject \( H_0 \). This implies that the regression relationship is significant.
Rejecting \( H_0 \) means the data provides sufficient evidence to infer that a linear relationship likely exists between the supply voltage and the current without electronics in this exercise. Understanding these concepts will help you evaluate regression models effectively.

Correlation Coefficient

The correlation coefficient, denoted by \( r \), is a statistic used to measure the strength and direction of a linear relationship between two variables. A value of \( r = 1 \) indicates a perfect positive correlation, \( -1 \) an exact negative correlation, and \( 0 \) no linear correlation.
In the context of this exercise, the correlation coefficient allows you to understand how changes in supply voltage relate to the current without electronics. The calculation of \( r \) involves several steps but leads to a straightforward interpretation:

• A positive \( r \) suggests that as one variable increases, so does the other.
• A negative \( r \) indicates an inverse relationship between the variables.

By computing \( r \), we can quantify the linear association and deduce if the supply voltage reliably predicts the current. This helps in confirming whether a linear regression model is appropriate for the data.

Confidence Interval

A confidence interval offers a range of values that likely contain the true parameter of interest, in this case, the correlation coefficient. It provides insight into the reliability of the \( r \) value calculated from data.
In our example, a 95% confidence interval means that if we repeated this study multiple times, 95% of the calculated intervals would include the true correlation coefficient.
The process uses Fisher's Z transformation to stabilize the variance of \( r \), making it easier to calculate the interval. After finding the interval in Z space, you convert back to the r scale, which allows you to better understand the possible range of the correlation. Using this interval, you determine whether zero is a plausible value for \( \rho \). If zero is not within the interval, it suggests a significant correlation. This method helps confirm evidence found through hypothesis testing, adding confidence to your conclusions.

Regression Statistics

Regression statistics are tools used to assess the quality of a regression model. Core statistics include the slope \( m \), intercept \( c \), and coefficient of determination \( R^2 \). Each reveals important aspects of your model:

• The slope indicates the rate of change of the dependent variable (current) with respect to the independent variable (voltage).
• The intercept represents the expected value of the dependent variable when the independent variable equals zero.
• The \( R^2 \) value shows the proportion of the variance in the dependent variable that is predictable from the independent variable. It ranges from 0 to 1, where values closer to 1 denote a better model fit.

By analyzing regression statistics in the context of the given data, you gain clarity on the data relationships. These statistics inform both the strength of the data relationships and the model's predictive capabilities, essential for understanding and developing precise regression models.