Two intervals are given, A and B, for the same value of the explanatory variable. A: 18.1 to 22.7 B: 16.6 to 24.2. What is the predicted value of the response variable for this value of the explanatory variable?
Question:Two intervals are given, A and B, for the same value of the explanatory variable. A: 18.1 to 22.7 B: 16.6 to 24.2. What is the predicted value of the response variable for this value of the explanatory variable?
Prediction Interval:The prediction intervals are used to estimate the range for the individual value of the dependent variable. Because it contains the individual values which have high variation as compared to the sample statistics i.e. sample mean, etc, so the width of the prediction interval is more than the interval calculated for the sample mean i.e. confidence interval.
Answer and Explanation:We have: Intervals A: <span id="MathJax-Element-1-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="(18.1,22.7)" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">(18.1,22.7)(18.1,22.7) Interval B: <span id="MathJax-Element-2-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="(16.6,24.2)" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">(16.6,24.2)(16.6,24.2)
The width of the interval B is more than the width of the interval A. Therefore, it can be said that the interval B is the prediction interval while the interval A is a confidence interval. It implies, to predict the value of the response variable we will use the interval B. The predicted value of the response variable for this value of the explanatory variable: <span id="MathJax-Element-3-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="=Mid value of the interval B=16.6+24.22=20.4" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">=Mid value of the interval B=16.6+24.22=20.4
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