Hurricanes Use the results of Problem 14 in Section 12.3 to an...

Question

Hurricanes Use the results of Problem 14 in Section 12.3 to answer the following questions:

b. Construct a 95% confidence interval for the mean wind speed found in part (a).

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A study finds the regression equation for predicting the weighty in units of kilograms of a fish from its length X in units of centimeters is Y is equal to 0.8 plus 0.05 multiplied by X. The data set has N equals 18, a mean length X bar is equal to 35. The sum of X subscript I minus X 2 is equal to 210. And standard error of estimate S is equal to 0.9. Construct a 99% prediction interval for a fish of length X subscript zero is equal to 40 centimeters. Awesome. So it appears for this particular prong we're asked to construct a 99% prediction interval for a fish of length X subscript zero is equal to 40 centimeters based on the information that is provided to us by the prom itself. So now that we know that we're ultimately trying to determine the specific interval value. Let's read off our multiple choice answers to see what our final answer might be, noting that they're all written in inside a parentheses because they're all written in interval notation. A is -2.157.4.315. B is 0.525.4.785. C is -0.050. 5.650 and D is 2.024.5.120. Fantastic. So our first step now is we need to take our regression equation and we need to plug in our value for X, which for this case we're told that the fish has a length of 40 centimeters. We're going to plug that value in for X into our regression equation and solve for Y hat. So Y hat is equal to 0.8 plus 0.05 multiplied by X, which in this case X is equal to 40. And when we plug in that expression into our calculator, we will find that it's simply equal to 2.8. Fantastic. Our second step now is we need to compute the value for S predicted, which I'll call SP for short, and SP predicted, as we should recall, how to use the equation and plugging in our known variables, we will note that SP is equal to 0.9. Multiplied by the square root of 1 + 1 divided by 18 plus parentheses 40 minus 35 2 divided by 210 and when we plug this expression into our calcul. Later and we round to 4 decimal places, we will find that it's approximately equal to 0.9754. So once again, SP is approximately equal to 0.9754. Fantastic. So now our third step now is we need to determine our degrees of freedom, which in this case will be N minus 2, which is equal to 16th. Which I'll call DF to denote degrees of freedom. And now that we know that our degrees of freedom is equal to 16, we need to note that for a 99% confidence, that will mean that our critical T value, which we will denote as T, will be approximately equal to 2.921. Fantastic. And using our critical T value, our fourth step is we need to determine our margin of error. So, our margin of error is equal to 2.921 multiplied by. Our value of 0.9754. Which is approximately equal to when we round to three decimal places, 2.850, fantastic. So now, our fifth and final step is we need to construct our prediction interval. So we need to take 2.8 plus or minus 2.850. And when we perform our prediction interval, we will find that it's equal to parentheses once again rounding to three decimal places because all of our multiple choice answers are rounded to three decimal places. We will find that our prediction interval yields an answer of parentheses negative 0.050. 5.650. And that's it. That's our final answer for the interval. Hooray, we did it, and this corresponds to multiple choice answer letter C, which once again is equal to parentheses 0.050. 5.650. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Hurricanes Use the results of Problem 14 in Section 12.3 to answer the following questions:
b. Construct a 95% confidence interval for the mean wind speed found in part (a).

Key Concepts

Confidence Interval

Sample Mean and Standard Deviation

t-Distribution and Degrees of Freedom

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