Question

Use the Excel output in the below table to do (1) through (6)
for each of*β*_{0}, *β*_{1},
*β*_{2}, and *β*_{3}*.*

y = *β*_{0} +
*β*_{1}*x*_{1} +
*β*_{2}*x*_{2} +
*β*_{3}*x*_{3} + ε
df = n – (k + 1) = 16 – (3 + 1) = 12

Excel output for the hospital labor needs case (sample size:
*n* = 16)

Coefficients | Standard Error | t Stat |
p-value |
Lower 95% | Upper 95% | |

Intercept | 1946.8020 | 504.1819 | 3.8613 | 0.0023 | 848.2840 | 3045.3201 |

XRay (x1) | 0.0386 | 0.0130 | 2.9579 | 0.0120 | 0.0102 | 0.0670 |

BedDays(x2) | 1.0394 | 0.0676 | 15.3857 | 2.91E-09 | 0.8922 | 1.1866 |

LengthSt(x3) | -413.7578 | 98.5983 | -4.1964 | 0.0012 | -628.5850 | -198.9306 |

**(1)** Find b_{j}, sb_{j}, and the
*t* statistic for testing *H*_{0}:
*β _{j}* = 0 on the output and report their values.

b_{j} |
sb_{j} |
t | |

H_{0}:
β_{0} = 0 |
|||

H_{0}:
β_{1} = 0 |
|||

H_{0}:
β_{2} = 0 |
|||

H_{0}:
β_{3} = 0 |
|||

**(2)** Using the *t* statistic and appropriate
critical values, test *H _{0}*:

t_{.025 ( ) }

H:
_{0}β_{0} =0; |
(Click to select)RejectDo not reject H |

H:
_{0}β_{1} =0; |
(Click to select)Do not rejectReject H |

H:
_{0}β_{2} =0; |
(Click to select)RejectDo not reject H |

H:
_{0}β_{3} =0; |
(Click to
select)RejectDo not reject H_{0} |

**(3)** Using the *t* statistic and appropriate
critical values, test *H _{0}*:

t_{.005 ( )}

H:
_{0}β_{0} =0; |
(Click to
select)Do not rejectReject H_{0} |

H:
_{0}β_{1} =0; |
(Click to
select)RejectDo not reject H_{0} |

H:
_{0}β_{2} =0; |
(Click to
select)Do not rejectReject H_{0} |

H:
_{0}β_{3} =0; |
(Click to
select)RejectDo not reject H_{0} |

**(4)** Find the *p*-value for testing
*H _{0}*:

H:
_{0}β_{1} = 0 is____ |
; Reject
Hat _{0}α = (Click to
select)0.010.050.001 |

H:
_{0}β_{3} = 0 is____ |
; Reject
Hat _{0}α = (Click to
select)0.050.010.001 |

**(5)** Calculate the 95 percent confidence interval
for *β _{j.}*

95% C.I. | |

β_{0} |
[, ] |

β_{1} |
[, ] |

β_{2} |
[, ] |

β_{3} |
[, ] |

Answer #1

given:

Coefficients | Standard Error | t Stat | p-value | Lower 95% | Upper 95% | |

Intercept | 1946.802 | 504.1819 | 3.8613 | 0.0023 | 848.284 | 3045.3201 |

XRay (x1) | 0.0386 | 0.013 | 2.9579 | 0.012 | 0.0102 | 0.067 |

BedDays(x2) | 1.0394 | 0.0676 | 15.3857 | 2.91E-09 | 0.8922 | 1.1866 |

LengthSt(x3) | -413.7578 | 98.5983 | -4.1964 | 0.0012 | -628.585 | -198.9306 |

1)

b_{j} |
sb_{j} |
t= bj/se_bj | |

H_{0}: β_{0} = 0 |
1946.8020 |
504.1819 |
3.861 |

H_{0}: β_{1} = 0 |
0.0386 |
0.0130 |
2.958 |

H_{0}: β_{2} = 0 |
1.0394 |
0.0676 |
15.386 |

H_{0}: β_{3} = 0 |
-413.7578 |
98.5983 |
-4.196 |

2)

alpha = 5% | ||

t(a/2,n-2) = t_{.025 (14)
} |
Decision (Reject Ho if |t|>t(a/2,n-2) ) | |

H_{0}:β_{0} =0; |
2.145 |
Reject Ho |

H_{0}: β_{1}=0; |
2.145 |
Reject Ho |

H_{0}: β_{2}=0; |
2.145 |
Reject Ho |

H_{0}: β_{3}=0; |
2.145 |
Reject Ho |

3)

alpha = 1% | ||

t(a/2,n-2) = t_{.005 (14)
} |
Decision (Reject Ho if |t|>t(a/2,n-2) ) | |

H_{0}:β_{0} =0; |
2.977 |
Reject Ho |

H_{0}: β_{1}=0; |
2.977 |
Do not reject Ho |

H_{0}: β_{2}=0; |
2.977 |
Reject Ho |

H_{0}: β_{3}=0; |
2.977 |
Reject Ho |

4)

P-value = 2*(1-P(T<|t|) with (n-2) d.f. | Decision (Reject Ho if p-value < alpha ) | ||

H_{0}:β_{0} =0; |
0.002 |
= T.DIST.2T(ABS(3.8613),16-2) | Reject H0 at α = (0.10, .05,
.01) |

H_{0}: β_{1}=0; |
0.010 |
= T.DIST.2T(ABS(2.9579),16-2) | Reject H0 at α = (0.10,
.05) |

H_{0}: β_{2}=0; |
0.0000 |
= T.DIST.2T(ABS(15.3857),16-2) | Reject H0 at α = (0.10, .05,
.01, and .001) |

H_{0}: β_{3}=0; |
0.0009 |
= T.DIST.2T(ABS(-4.1964),16-2) | Reject H0 at α = (0.10, .05,
.01, and .001) |

5)

95% C.I. | Lower = bj - t(a/2,n-2)*SE_bj | Upper = bj + t(a/2,n-2)*SE_bj | ||

β_{0} |
865.332 |
1946.802-2.145*504.1819 | 3028.272 |
1946.802+2.145*504.1819 |

β_{1} |
0.011 |
0.0386-2.145*0.013 | 0.066 |
0.0386+2.145*0.013 |

β_{2} |
0.894 |
1.0394-2.145*0.0676 | 1.184 |
1.0394+2.145*0.0676 |

β_{3} |
-625.251 |
-413.7578-2.145*98.5983 | -202.264 |
-413.7578+2.145*98.5983 |

Use the Excel output in the below table to do (1) through (6)
for each ofβ0, β1,
β2, and β3.
y = β0 +
β1x1 +
β2x2 +
β3x3 + ε
df = n – (k + 1) = 16 – (3 + 1) = 12
Excel output for the hospital labor needs case (sample size:
n = 16)
Coefficients
Standard
Error
t Stat
p-value
Lower 95%
Upper 95%
Intercept
1946.8020
504.1819
3.8613
0.0023
848.2840
3045.3201
XRay (x1)
0.0386...

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to perform the regression analysis and the t statistic for
testing H0: β1 = 0 versus
Ha: β1 ≠ 0.
ANOVA
df
SS
MS
F
Significance F
Regression
1
61,091.6455
61,091.6455
.69
.4259
Residual
10
886,599.2711
88,659.9271
Total
11
947,690.9167
(n = 12;...

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