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Chapter 13: Problem 34

Explain the difference between a confidence interval and a prediction interval. How can a prediction level of \(95 \%\) be interpreted?

Short Answer

Expert verified
A confidence interval estimates the unknown population parameter while a prediction interval estimates future observations. Both provide a range of plausible values, but for different quantities of interest. A 95% prediction level means future observations will fall within the prediction interval 95% of the time.

Step by step solution

01

Defining Confidence Interval

A confidence interval is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. It gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.
02

Defining Prediction Interval

A prediction interval is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Unlike confidence intervals that predict parameters of population, prediction intervals predict individual data points.
03

Differentiating Between Confidence and Prediction Intervals

While both confidence and prediction intervals provide a range of plausible values for certain quantities of interest, they are not the same. The former deals with unknown population parameters and the latter with future observation.
04

Interpreting a 95% prediction level

A 95% prediction level can be interpreted as meaning that future observations will fall within the prediction interval 95% of the time. It's a range where individual data points are likely to fall, with 95% certainty, given what has already been observed.

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Most popular questions from this chapter

An investigation of the relationship between traf. fic flow \(x\) (thousands of cars per \(24 \mathrm{hr}\) ) and lead content \(y\) of bark on trees near the highway (mg/g dry weight) yielded the accompanying data. A simple linear regression model was fit, and the resulting estimated regression line was \(\hat{y}=28.7+33.3 x .\) Both residuals and standardized residuals are also given. a. Plot the \((x\), residual \()\) pairs. Does the resulting plot suggest that a simple linear regression model is an appropriate choice? Explain your reasoning. b. Construct a standardized residual plot. Does the plot differ significantly in general appearance from the plot in Part (a)?

The data of Exercise \(13.25\), in which \(x=\) milk temperature and \(y=\) milk \(\mathrm{pH}\), yield $$ \begin{aligned} &n=16 \quad \bar{x}=43.375 \quad S_{x x}=7325.75 \\ &b=-.00730608 \quad a=6.843345 \quad s_{e}=.0356 \end{aligned} $$ a. Obtain a \(95 \%\) confidence interval for \(\alpha+\beta(40)\), the true average milk \(\mathrm{pH}\) when the milk temperature is \(40^{\circ} \mathrm{C}\). b. Calculate a \(99 \%\) confidence interval for the true average milk pH when the milk temperature is \(35^{\circ} \mathrm{C}\). c. Would you recommend using the data to calculate a \(95 \%\) confidence interval for the true average \(\mathrm{pH}\) when the temperature is \(90^{\circ} \mathrm{C}\) ? Why or why not?

The flow rate in a device used for air quality measurement depends on the pressure drop \(x\) (inches of water) across the device's filter. Suppose that for \(x\) values between 5 and 20 , these two variables are related according to the simple linear regression model with true regression line \(y=-0.12+0.095 x\). a. What is the true average flow rate for a pressure drop of 10 in.? A drop of 15 in.? b. What is the true average change in flow rate associated with a 1 -in. increase in pressure drop? Explain. c. What is the average change in flow rate when pressure drop decreases by 5 in.?

Suppose that a regression data set is given and you are asked to obtain a confidence interval. How would you tell from the phrasing of the request whether the interval is for \(\beta\) or for \(\alpha+\beta x^{*} ?\)

The article "Cost-Effectiveness in Public Education" (Chance [1995]: \(38-41\) ) reported that, for a sample of \(n=44\) New Jersey school districts, a regression of \(y=\) average SAT score on \(x=\) expenditure per pupil (thousands of dollars) gave \(b=15.0\) and \(s_{b}=5.3\). a. Does the simple linear regression model specify a useful relationship between \(x\) and \(y\) ? b. Calculate and interpret a confidence interval for \(\beta\) based on a \(95 \%\) confidence level.
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