HP 48gII - Manual (Page 624)

Page 18-53
• Hypothesis testing on the intercept , Α:
Null hypothesis, H
0
: Α = Α
0
, tested against the alternative hypothesis, H
1
:
Α ≠ Α
0
. The test statistic is t
0
= (a- Α
0
)/[(1/n)+ x 2 /S
xx
] 1/2 , where t follows
the Student’s t distribution with ν = n – 2, degrees of freedom, and n
represents the number of points in the sample. The test is carried out as
that of a mean value hypothesis testing, i.e., given the level of
significance,
α, determine the critical value of t, t
α /2
, then, reject H
0
if t
0
>
t
α /2
or if t
0
< - t
α /2
.
• Confidence interval for the mean value of Y at x = x
0
, i.e., α+βx
0
:
a+b ⋅x−(t
n-2, α /2
) ⋅s
e
⋅[(1/n)+(x
0
- x) 2 /S
xx
] 1/2 < α+βx
0
<
a+b ⋅x+(t
n-2, α /2
) ⋅s
e
⋅[(1/n)+(x
0
- x) 2 /S
xx
] 1/2 .
• Limits of prediction: confidence interval for the predicted value Y
0
=Y(x
0
):
a+b ⋅x−(t
n-2, α /2
) ⋅s
e
⋅[1+(1/n)+(x
0
- x) 2 /S
xx
] 1/2 < Y
0
<
a+b ⋅x+(t
n-2, α /2
) ⋅s
e
⋅[1+(1/n)+(x
0
- x) 2 /S
xx
] 1/2 .
Procedure for inference statistics for linear regression using the
calculator
1) Enter (x,y) as columns of data in the statistical matrix
ΣDAT.
2) Produce a scatterplot for the appropriate columns of ΣDAT, and use
appropriate H- and V-VIEWS to check linear trend.
3) Use ‚Ù˜˜@@@OK@@@, to fit straight line, and get a, b, s
xy
(Covariance), and r
xy
(Correlation).
4) Use ‚Ù˜@@@OK@@@, to obtain x, y, s
x
, s
y
. Column 1 will show the
statistics for x while column 2 will show the statistics for y.
5) Calculate
2 ) 1 (
x xx
s n S ⋅ − =
,
) 1 (
2
1
2 2 2
xy y e
r s
n
n
s − ⋅ ⋅
−
−
=
6) For either confidence intervals or two-tailed tests, obtain t
α /2
, with (1-
α)100% confidence, from t-distribution with ν = n -2.
7) For one- or two-tailed tests, find the value of t using the appropriate
equation for either Α or Β. Reject the null hypothesis if P-value < α.
8) For confidence intervals use the appropriate formulas as shown above.