Question

data: n=22
13.9 10.9 9.3 5.2 14.0 2.0 3.2 12.0 13.0 11.3 12.2
5.0 2.0 2.0 17.1 13.4 6.0 14.0 18.0 13.6 5.4 8.7

1. compute x and s²;
2. test H0 : u = 4.50 vs. HA : u is not= 4.50 at the 5% significance level.
3. construct a 95% confidence interval for u.
4. construct a 99% confidence interval for u.
5. report your p-value of the test in (2).

Answer 1

	Values ( X )	Σ ( Xi- X̅ )2
	13.9	18.1008
	10.9	1.5738
	9.3	0.1194
	5.2	19.7625
	14	18.9617
	2	58.4537
	3.2	41.5445
	12	5.5437
	13	11.2527
	11.3	2.7374
	12.2	6.5255
	5	21.5807
	2	58.4537
	2	58.4537
	17.1	55.5696
	13.4	14.0963
	6	13.2897
	14	18.9617
	18	69.7977
	13.6	15.6381
	5.4	18.0243
	8.7	0.894
Total	212.2	529.3352

Part 1)

Mean X̅ = Σ Xi / n
X̅ = 212.2 / 22 = 9.6455
Sample Standard deviation S_X = √ ( (Xi - X̅ )² / n - 1 )
S_X = √ ( 529.3352 / 22 -1 ) = 5.0206
S² = 25.2064

Part 2)

To Test :-
H0 :- µ = 4.50
H1 :- µ ≠ 4.50

Test Statistic :-
t = ( X̅ - µ ) / ( S / √(n))
t = ( 9.6455 - 4.5 ) / ( 5.0206 / √(22) )
t = 4.8071

Test Criteria :-
Reject null hypothesis if | t | > t(α/2, n-1)
Critical value t(α/2, n-1) = t(0.05 /2, 22-1) = 2.08
| t | > t(α/2, n-1) = 4.8071 > 2.08
Result :- Reject null hypothesis

Part 3)

Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
Critical value t(α/2, n-1) = t(0.05 /2, 22- 1 ) = 2.08 ( From t table )
9.6455 ± t(0.05/2, 22 -1) * 5.0206/√(22)
Lower Limit = 9.6455 - t(0.05/2, 22 -1) 5.0206/√(22)
Lower Limit = 7.4191
Upper Limit = 9.6455 + t(0.05/2, 22 -1) 5.0206/√(22)
Upper Limit = 11.8719
95% Confidence interval is ( 7.4191 , 11.8719 )

Part 4)

Confidence Interval
X̅ ± t(α/2, n-1) S/√(n)
Critical value t(α/2, n-1) = t(0.01 /2, 22- 1 ) = 2.831 ( From t table )
9.6455 ± t(0.01/2, 22 -1) * 5.0206/√(22)
Lower Limit = 9.6455 - t(0.01/2, 22 -1) 5.0206/√(22)
Lower Limit = 6.6152
Upper Limit = 9.6455 + t(0.01/2, 22 -1) 5.0206/√(22)
Upper Limit = 12.6758
99% Confidence interval is ( 6.6152 , 12.6758 )

Part 5)

P - value = P ( t > 4.8071 ) = 0.0001
Looking for the value t = t = 4.8071 in t table across n - 1 = 21 degree of freedom.