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Eruptions of the Old Faithful geyser in Yellowstone
Chapter 7, Problem 3SE(choose chapter or problem)
Eruptions of the Old Faithful geyser in Yellowstone National Park typically last from 1.5 to 5 minutes. Between eruptions are dormant periods, which typically last from 50 to 100 minutes. A dormant period can also be thought of as the waiting time between eruptions. The durations in minutes for 60 consecutive dormant periods are given in the following table. It is desired to predict the length of a dormant period from the length of the dormant period immediately preceding it. To express this in symbols, denote the sequence of dormant periods \(T_1\), . . . , \(T_{60}\). It is desired to predict \(T_{i+1}\) from \(T_i\).
\(\begin{array}{rc||cc||cc||cc||cc||cc}
\hline \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} & \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} & \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} & \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} & \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} & \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} \\
\hline 1 & 80 & 11 & 56 & 21 & 82 & 31 & 88 & 41 & 72 & 51 & 67 \\
2 & 84 & 12 & 80 & 22 & 51 & 32 & 51 & 42 & 75 & 52 & 81 \\
3 & 50 & 13 & 69 & 23 & 76 & 33 & 80 & 43 & 75 & 53 & 76 \\
4 & 93 & 14 & 57 & 24 & 82 & 34 & 49 & 44 & 66 & 54 & 83 \\
5 & 55 & 15 & 90 & 25 & 84 & 35 & 82 & 45 & 84 & 55 & 76 \\
6 & 76 & 16 & 42 & 26 & 53 & 36 & 75 & 46 & 70 & 56 & 55 \\
7 & 58 & 17 & 91 & 27 & 86 & 37 & 73 & 47 & 79 & 57 & 73 \\
8 & 74 & 18 & 51 & 28 & 51 & 38 & 67 & 48 & 60 & 58 & 56 \\
9 & 75 & 19 & 79 & 29 & 85 & 39 & 68 & 49 & 86 & 59 & 83 \\
10 & 80 & 20 & 53 & 30 & 45 & 40 & 86 & 50 & 71 & 60 & 57 \\
\hline
\end{array}\)
a. Construct a scatterplot of the points \((T_i,T_{i+1})\), for \(i=1\), . . . , 59.
b. Compute the least-squares line for predicting \(T_{i+1}\) from \(T_i\). (Hint: The values of the independent variable \((x)\) are \(T_{1}\), . . . , \(T_{59}\), and the values of the dependent variable \((y)\) are \(T_2\), . . . , \(T_{60}\).)
c. Find a 95% confidence interval for the slope \(\beta_1\).
d. If the waiting time before the last eruption was 70 minutes, what is the predicted waiting time before the next eruption?
e. Find a 98% confidence interval for the mean waiting time before the next eruption when the time before the last eruption was 70 minutes.
f. Find a 99% prediction interval for the waiting time before the next eruption, if the time before the last eruption was 70 minutes.
Questions & Answers
QUESTION:
Eruptions of the Old Faithful geyser in Yellowstone National Park typically last from 1.5 to 5 minutes. Between eruptions are dormant periods, which typically last from 50 to 100 minutes. A dormant period can also be thought of as the waiting time between eruptions. The durations in minutes for 60 consecutive dormant periods are given in the following table. It is desired to predict the length of a dormant period from the length of the dormant period immediately preceding it. To express this in symbols, denote the sequence of dormant periods \(T_1\), . . . , \(T_{60}\). It is desired to predict \(T_{i+1}\) from \(T_i\).
\(\begin{array}{rc||cc||cc||cc||cc||cc}
\hline \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} & \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} & \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} & \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} & \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} & \boldsymbol{i} & \boldsymbol{T}_{\boldsymbol{i}} \\
\hline 1 & 80 & 11 & 56 & 21 & 82 & 31 & 88 & 41 & 72 & 51 & 67 \\
2 & 84 & 12 & 80 & 22 & 51 & 32 & 51 & 42 & 75 & 52 & 81 \\
3 & 50 & 13 & 69 & 23 & 76 & 33 & 80 & 43 & 75 & 53 & 76 \\
4 & 93 & 14 & 57 & 24 & 82 & 34 & 49 & 44 & 66 & 54 & 83 \\
5 & 55 & 15 & 90 & 25 & 84 & 35 & 82 & 45 & 84 & 55 & 76 \\
6 & 76 & 16 & 42 & 26 & 53 & 36 & 75 & 46 & 70 & 56 & 55 \\
7 & 58 & 17 & 91 & 27 & 86 & 37 & 73 & 47 & 79 & 57 & 73 \\
8 & 74 & 18 & 51 & 28 & 51 & 38 & 67 & 48 & 60 & 58 & 56 \\
9 & 75 & 19 & 79 & 29 & 85 & 39 & 68 & 49 & 86 & 59 & 83 \\
10 & 80 & 20 & 53 & 30 & 45 & 40 & 86 & 50 & 71 & 60 & 57 \\
\hline
\end{array}\)
a. Construct a scatterplot of the points \((T_i,T_{i+1})\), for \(i=1\), . . . , 59.
b. Compute the least-squares line for predicting \(T_{i+1}\) from \(T_i\). (Hint: The values of the independent variable \((x)\) are \(T_{1}\), . . . , \(T_{59}\), and the values of the dependent variable \((y)\) are \(T_2\), . . . , \(T_{60}\).)
c. Find a 95% confidence interval for the slope \(\beta_1\).
d. If the waiting time before the last eruption was 70 minutes, what is the predicted waiting time before the next eruption?
e. Find a 98% confidence interval for the mean waiting time before the next eruption when the time before the last eruption was 70 minutes.
f. Find a 99% prediction interval for the waiting time before the next eruption, if the time before the last eruption was 70 minutes.
ANSWER:Step 1 of 8
Simple Linear Regression is a statistical method used to model and analyze the relationship between two quantitative variables. In this regression type, a linear relationship is sought between a dependent variable (also termed the response variable) and an independent variable (also referred to as the predictor variable). The objective is to employ the independent variable for predicting or explaining the variation observed in the dependent variable.