A chemical reaction is run 12 times, and the temperature xi (in °C) and the yield yi (in percent of a theoretical maximum) is recorded each time. The following summary statistics are recorded: x⎯⎯=65.0, y⎯⎯=29.03,∑ni=1(xi−x⎯⎯)2=6032.0,∑ni=1(yi−y⎯⎯)2=835.42,∑ni=1(xi−x⎯⎯)(yi−y⎯⎯)=1988.5x¯=65.0, y¯=29.03,∑i=1n(xi−x¯)2=6032.0,∑i=1n(yi−y¯)2=835.42,∑i=1n(xi−x¯)(yi−y¯)=1988.5 Let β0 represent the hypothetical yield at a temperature of 0°C, and let β1 represent the increase in yield caused by an increase in temperature of 1°C. Assume that assumptions 1 through 4 for errors in linear models hold. Find 95% confidence intervals for β0 and β1. Round the answers to three decimal places. The 95% confidence interval for β0 is ( , ). The 95% confidence interval for β1 is ( , ).
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
A chemical reaction is run 12 times, and the temperature xi (in °C) and the yield yi (in percent of a theoretical maximum) is recorded each time. The following summary statistics are recorded:
x⎯⎯=65.0, y⎯⎯=29.03,∑ni=1(xi−x⎯⎯)2=6032.0,∑ni=1(yi−y⎯⎯)2=835.42,∑ni=1(xi−x⎯⎯)(yi−y⎯⎯)=1988.5x¯=65.0, y¯=29.03,∑i=1n(xi−x¯)2=6032.0,∑i=1n(yi−y¯)2=835.42,∑i=1n(xi−x¯)(yi−y¯)=1988.5
Let β0 represent the hypothetical yield at a temperature of 0°C, and let β1 represent the increase in yield caused by an increase in temperature of 1°C. Assume that assumptions 1 through 4 for errors in linear models hold.
Find 95% confidence intervals for β0 and β1. Round the answers to three decimal places.
The 95% confidence interval for β0 is ( , ).
The 95% confidence interval for β1 is ( , ).
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