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Chapter 15: Problem 14

A journalist is reporting about some research on appropriate amounts of sleep for people 9 to 19 years of age. In that research, a linear regression model is used to describe the relationship between alertness and number of hours of sleep the night before. The researchers reported a \(95 \%\) confidence interval, but newspapers usually report an estimate and a margin of error. Explain how the journalist could determine the margin of error from the reported confidence interval.

Short Answer

Expert verified
To find the margin of error from the reported confidence interval, the journalist should first identify the lower bound (a) and upper bound (b) of the given confidence interval. Then, calculate the midpoint (estimate) as \(\frac{a+b}{2}\), and calculate the margin of error as \(\frac{b - a}{2}\). Finally, report the estimate and margin of error as "Estimate: Midpoint 卤 Margin of Error".

Step by step solution

01

Understand confidence interval and margin of error

A confidence interval is a range of values that estimates a population parameter with a specified level of confidence (in this case, 95%). It is usually represented as `(Lower Bound, Upper Bound)`. The margin of error is the measurement of uncertainty when estimating the true population value. It is calculated as half of the range within the confidence interval.
02

Extract the lower and upper bound from the given confidence interval

Suppose the researchers reported a 95% confidence interval like `(a, b)`, where `a` represents the lower bound, and `b` represents the upper bound.
03

Calculate the midpoint of the confidence interval

To do this, we will need to find the average of the lower and upper bounds. That is: \(Midpoint = \frac{a+b}{2}\)
04

Calculate the margin of error

Now, we need to calculate the margin of error. Since the margin of error is half of the range within the confidence interval, we can calculate it as follows: \(Margin\ of\ Error = \frac{b - a}{2}\)
05

Report the estimate and margin of error

Using the midpoint as the estimate and the margin of error calculated in step 4, the journalist can report the relationship between alertness and the number of hours slept as: "Estimate: Midpoint 卤 Margin of Error"

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Key Concepts

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

Margin of Error
When you're trying to figure out what a survey or study is really saying, it's important to consider the margin of error. This term represents how much you can expect your finding to vary from the true population value. Think of it as the wiggle room in a scientific estimate. To calculate the margin of error, look at the reported confidence interval, often written like this: (Lower Bound, Upper Bound). The margin of error is simply half of the range of this interval:\[Margin\ of\ Error = \frac{b - a}{2}\]This tells us how far off the estimate might be from the true population value, providing a clearer picture of the data's accuracy. If you see a large margin of error, it means there's more uncertainty in the estimate.
Linear Regression
Linear regression is a powerful statistical tool used to understand the relationship between two variables. In this exercise, it helps describe the link between alertness and hours of sleep. Essentially, it tries to "fit a line" through the data points that best explains how these two elements are connected.
  • The independent variable: This is what you think is impacting the other. Here, it's the number of hours of sleep.
  • The dependent variable: This is what you measure to see if it changes because of the independent variable. In this case, that鈥檚 alertness.
By plotting these on a graph and using the line, linear regression predicts the dependent variable based on the independent one. The more closely aligned the data points are to this line, the stronger the relationship.
Population Parameter Estimation
In statistics, when you hear about estimating a population parameter, it means we're trying to make an educated guess about an entire group's characteristic based on a sample. This is crucial in research because it's often impractical to study a whole population directly.
  • Parameters: These are the actual values we're interested in that describe the population, like the average hours of sleep needed.
  • Statistics: Values that describe the sample. For example, the average alertness level from those sampled.
By using confidence intervals around these sample statistics, we can provide an estimate for the population parameter, in which we state the likely range (the confidence interval) where the parameter lies, based on our sample.

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Most popular questions from this chapter

Let \(x\) be the size of a house (in square feet) and \(y\) be the amount of natural gas used (therms) during a specified period. Suppose that for a particular community, \(x\) and \(y\) are related according to the simple linear regression model with \(\beta=\) slope of population regression line \(=.017\) \(\alpha=y\) intercept of population regression line \(=-5.0\) Houses in this community range in size from 1000 to 3000 square feet. a. What is the equation of the population regression line? b. Graph the population regression line by first finding the point on the line corresponding to \(x=1000\) and then the point corresponding to \(x=2000\), and drawing a line through these points. c. What is the mean value of gas usage for houses with 2100 sq. ft. of space? d. What is the average change in usage associated with a 1 sq. ft. increase in size? e. What is the average change in usage associated with a 100 sq. ft. increase in size? f. Would you use the model to predict mean usage for a 500 sq. ft. house? Why or why not?

Explain what distinguishes a deterministic model from a probabilistic model.

The SAT and ACT exams are often used to predict a student's first-term college grade point average (GPA). Different formulas are used for different colleges and majors. Suppose that a student is applying to State U with an intended major in civil engineering. Also suppose that for this college and this major, the following model is used to predict first term GPA. $$ \begin{aligned} G P A &=a+b(A C T) \\ a &=0.5 \\ b &=0.1 \end{aligned} $$ a. In this context, what would be the appropriate interpretation of the value of \(a\) ? b. In this context, what would be the appropriate interpretation of the value of \(b ?\)

The paper "Predicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thickness, Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors" (International journal of Poultry Science [2008]: \(85-88\) ) suggests that the simple linear regression model is reasonable for describing the relationship between \(y=\) eggshell thickness (in micrometers) and \(x=\) egg length (mm) for quail eggs. Suppose that the population regression line is \(y=0.135+0.003 x\) and that \(\sigma=0.005 .\) Then, for a fixed \(x\) value, \(y\) has a normal distribution with mean \(0.135+0.003 x\) and standard deviation 0.005 . a. What is the mean eggshell thickness for quail eggs that are \(15 \mathrm{~mm}\) in length? For quail eggs that are \(17 \mathrm{~mm}\) in length? b. What is the probability that a quail egg with a length of \(15 \mathrm{~mm}\) will have a shell thickness that is greater than \(0.18 \mu \mathrm{m} ?\) c. Approximately what proportion of quail eggs of length \(14 \mathrm{~mm}\) have a shell thickness of greater than \(0.175 ?\) Less than \(0.178 ?\)

The article "Vital Dimensions in Volume Perception: Can the Eye Fool the Stomach?" (Journal of Marketing Research [1999]: \(313-326\) ) gave the accompanying data on the dimensions (in \(\mathrm{cm}\) ) of the containers for 27 representative food products (Gerber baby food, Cheez Whiz, Skippy Peanut Butter, and Ahmed's tandoori paste, to name a few). a. Fit the simple linear regression model that would allow prediction of the maximum width of a food container based on its minimum width. b. Calculate the standardized residuals (or just the residuals if you don't have access to a computer program that gives standardized residuals) and make a residual plot to determine whether there are any outliers. c. The data point with the largest residual is for a 1 -liter Coke bottle. Delete this data point and refit the regression. Did deletion of this point result in a large change in the equation of the estimated regression line? d. For the regression line of Part (c), interpret the estimated slope and, if appropriate, the estimated intercept. e. For the data set with the Coke bottle deleted, do you think that the assumptions of the simple linear regression model are reasonable? Give statistical evidence for your answer.
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