Consider the following time series.

t yt t yt t yt t yt t yt

$16620112816382148$

$212720122917392251$

$39821133218432351$

$414924143319442455$

$5151024153620462557$

$($a$)$ Choose the correct time series plot.

$($i$)$

The time series plot represents trend. The horizontal axis is labeled t with values from $0$ to $25.$ The vertical axis is labeled y of t with values from $0$ to $60.$ The chart has a collection of $25$ points connected by line segments. The segments and the approximate points they connect are as follows. The segments start at $(1,6),$ go up and right to $(2,12),$ go down and right to $(3,9),$ go up and right to $(4,14),$ go up and right to $(5,15),$ go up and right to $(6,20),$ go right to $(7,20),$ go up and right to $(8,21),$ go up and right to $(9,24),$ go right to $(10,24),$ go up and right to $(11,28),$ go up and right to $(12,29),$ go up and right to $(13,32),$ go up and right to $(14,33),$ go up and right to $(15,36),$ go up and right to $(16,38),$ go up and right to $(17,39),$ go up and right to $(18,43),$ go up and right to $(19,44),$ go up and right to $(20,46),$ go up and right to $(21,48),$ go up and right to $(22,51),$ go right to $(23,51),$ go up and right to $(24,55),$ and go up and right to stop at $(25,57).($ii$)$

The time series plot represents trend. The horizontal axis is labeled t with values from $0$ to $25.$ The vertical axis is labeled y of t with values from $0$ to $60.$ The chart has a collection of $25$ points connected by line segments. The segments and the approximate points they connect are as follows. The segments start at $(1,55),$ go up and right to $(2,57),$ go down and right to $(3,51),$ go right to $(4,51),$ go down and right to $(5,46),$ go up and right to $(6,48),$ go down and right to $(7,43),$ go up and right to $(8,44),$ go down and right to $(9,38),$ go up and right to $(10,39),$ go down and right to $(11,33),$ go up and right to $(12,36),$ go down and right to $(13,29),$ go up and right to $(14,32),$ go down and right to $(15,24),$ go up and right to $(16,28),$ go down and right to $(17,21),$ go up and right to $(18,24),$ go down and right to $(19,20),$ go right to $(20,20),$ go down and right to $(21,14),$ go up and right to $(22,15),$ go down and right to $(23,9),$ go up and right to $(24,12),$ and go down and right to stop at $(25,6).$

$($iii$)$

The time series plot represents trend. The horizontal axis is labeled t with values from $0$ to $25.$ The vertical axis is labeled y of t with values from $0$ to $60.$ The chart has a collection of $25$ points connected by line segments. The segments and the approximate points they connect are as follows. The segments start at $(1,12),$ go down and right to $(2,6),$ go up and right to $(3,14),$ go down and right to $(4,9),$ go up and right to $(5,20),$ go down and right to $(6,15),$ go up and right to $(7,21),$ go down and right to $(8,20),$ go up and right to $(9,24),$ go right to $(10,24),$ go up and right to $(11,29),$ go down and right to $(12,28),$ go up and right to $(13,33),$ go down and right to $(14,32),$ go up and right to $(15,38),$ go down and right to $(16,36),$ go up and right to $(17,43),$ go down and right to $(18,39),$ go up and right to $(19,46),$ go down and right to $(20,44),$ go up and right to $(21,51),$ go down and right to $(22,48),$ go up and right to $(23,55),$ go down and right to $(24,51),$ and go up and right to stop at $(25,57).($iv$)$

The time series plot represents trend. The horizontal axis is labeled t with values from $0$ to $25.$ The vertical axis is labeled y of t with values from $0$ to $60.$ The chart has a collection of $25$ points connected by line segments. The segments and the approximate points they connect are as follows. The segments start at $(1,57),$ go down and right to $(2,55),$ go down and right to $(3,51),$ go right to $(4,51),$ go down and right to $(5,48),$ go down and right to $(6,46),$ go down and right to $(7,44),$ go down and right to $(8,43),$ go down and right to $(9,39),$ go down and right to $(10,38),$ go down and right to $(11,36),$ go down and right to $(12,33),$ go down and right to $(13,32),$ go down and right to $(14,29),$ go down and right to $(15,28),$ go down and right to $(16,24),$ go right to $(17,24),$ go down and right to $(18,21),$ go down and right to $(19,20),$ go right to $(20,20),$ go down and right to $(21,15),$ go down and right to $(22,14),$ go down and right to $(23,12),$ go down and right to $(24,9),$ and go down and right to stop at $(25,6).$

Plot $($i$)$

What type of pattern exists in the data?

The time series plot shows a

positive

linear

trend.

$($b$)$ Use simple linear regression analysis to find the parameters for the line that minimizes MSE for this time series.

If required, round your answers to three decimal places.

y$-$intercept, b$0=$

$-18\mathrm{.}000$

Slope, b$1=$

$3\mathrm{.}532$

MSE $=$

$136\mathrm{.}765$

$($c$)$ What is the forecast for t $=26?$

If required, round your answer to two decimal places.

$73\mathrm{.}84$