Mathematics, 10.12.2020 16:10 manarsadi6

Consider the following data on x = rainfall volume (m^3) and y = runoff volume (m^3) for a particular location. x 4 12 14 20 23 30 40 48 55 67 72 85 96 112 127
y 4 10 13 15 15 25 27 44 38 46 53 71 82 99 101

Required:
Calculate a 95% confidence interval for the true average change in runoff volume associated with a 1m^3 increase in rainfall volume. (Round your answers to three decimal places.)

Answers: 2

Show answers

Answers

Answer from: 4804053670

Step-by-step explanation:

From the given information:

x y xy x² y²

4 4 16 16 16

12 10 120 144 100

14 13 182 196 169

20 15 300 400 225

23 15 345 529 225

30 25 750 900 625

40 27 1080 1600 729

48 44 2112 2304 1936

55 38 2090 3025 1444

67 46 3082 4489 2116

72 53 3816 5184 2809

85 71 6035 7225 5041

96 82 7872 9216 6724

112 99 11088 12544 9801

127 101 12827 16129 10201

$\sum _{xi} = 805$ $\sum_{yi} = 643$ $\sum_{x_iy_I}= 51715$ $\sum_{x_i^2}= 63901$ $\sum_{y_i^2} = 42161$

The least-square regression equation is: $\hat y = b_o+b_1 x$

$b_1 = \dfrac{n \sum xy - ( \sum _x) ( \sum_y)}{n \sum x^2 - ( \sum x)^2}$

$b_1 = \dfrac{15(51715) - (803) (643)}{15(63901)-(805)^2}$

$b_1 = \dfrac{775725-516329}{958515-648025}$

$b_1 = \dfrac{259396}{310490}$

b₁ = 0.835440

∴ Slope term, b₁ = 0.835

$SS_{XX} = \sum x^2 - \dfrac{(\sum x)^2}{n}= 63901 - \dfrac{(805)^2}{15}=20699.33$

$SS_{yy} = \sum y^2 - \dfrac{(\sum y)^2}{n }= 42161 - \dfrac{(643)^2}{15}= 14597.73$

$SS_{xy} = \sum xy - \dfrac{(\sum x) (\sum y)}{n}= 51715 - \dfrac{(803)(643)}{15}= 17293.067$

$SST = SS_{yy}= 14597.73$

$SSR = \dfrac{SS_{xy}^2}{SS_{xx}}= \dfrac{17293.067^2}{20699.33}=14447.34$

SSE = SST - SSR = 14597.73 - 14447.34 = 150.39

The hypothesis test for the significance of $\beta_1$ is:

$H_o: \beta_1 = 0 \\ \\ H_1: \beta_1 \ne 0$

Significance level ∝ = 1 - 0.95 = 0.05

The sample slope b_1 = 0.835440

$Test \ statistic = t_{observed} = \dfrac{b_1-0}{\sqrt{\dfrac{SSE}{(n-2)SS_{xx}}}}$

$t_{o} = \dfrac{0.835440-0}{\sqrt{\dfrac{150.39}{(15-2)20699.33}}}$

$t_{o} = \dfrac{0.835440}{\sqrt{\dfrac{150.39}{269091.29}}}$

t_o = 35.339

Degree of freedom df = n - 2

df = 15 -2

df = 13

Using the Excel formula to determine the P_value.

$P-value = P(t, \Big|35.339 \Big|)$

P-value = 2 × t.dist(35.339,13,1)

P-value = 0.0000

P-value = 0

Critical value: $t_{critical} = t_{\alpha/2,df} = t_{0.05/2,13}= 2.160$

Rejection region: To reject H_o ; if $\Big | t_o \Big | t_c$

Decision: Since $\Big | t_o \Big | t_c$ ; we reject H_o

Conclusion: There is enough evidence to conclude that the linear relationship between x & y

Thus; we reject H_o & there is a useful linear relationship between x & y.

The 95% C.I for slope is given by the equation:

$=b_1 \pm t_{(\alpha/2,n-2)} \sqrt{\dfrac{SSE}{n-2}}\sqrt{\dfrac{1}{SS_{xx}} }$

$=0.835440 \pm 2.160 \sqrt{\dfrac{150.39}{15-2}}\sqrt{\dfrac{1}{20699.33} }$

= 0.835440 ± 2.160 (3.40124)(0.006951)

= 0.835440 ± 0.0511

= (0.835440 - 0.0511, 0.835440 + 0.0511)

= (0.78434, 0.88654)

= (0.784, 0.887) to three decimal places.

∴ 95% C.I of slope = $\mathbf{( 0.784 < \beta_1 < 0.887) \ to \ 3 \ d.p}$

Answer from: Quest

345

step-by-step explanation:

Answer from: Quest

Ralph is earning 2.8% commission

Another question on Mathematics

Mathematics, 21.06.2019 19:00

The test scores of 32 students are listed below. construct a boxplot for the data set and include the values of the 5-number summary. 32 37 41 44 46 48 53 55 57 57 59 63 65 66 68 69 70 71 74 74 75 77 78 79 81 82 83 86 89 92 95 99

Answers: 1

Answer

Mathematics, 21.06.2019 23:50

Write the standard form of an equation with (see picture below)

Answers: 2

Answer

Mathematics, 22.06.2019 03:30

Graph the linear equation find three points on the graph -x+2y=11

Answers: 2

Answer

Mathematics, 22.06.2019 04:00

Maria has $10,000 to invest in two accounts.he decides to invest one part in an account with 5% interest and the rest in an account with 8% interest. at the end of the year he has earned $702,50 in interest.¿how much did maria invest in the 8% account?

Answers: 2

Answer

You know the right answer?

Consider the following data on x = rainfall volume (m^3) and y = runoff volume (m^3) for a particula...

Questions

Mathematics, 05.11.2020 02:00

PLS HELP! Mrs. Duzwell bought a box of 150 pencils for $12. a.) Let p equal the unit price per pencil. Write a multiplication equation that rep...