Hypothesis Testing and Standard Normal Distribution

This article covers topics such as two-tailed and one-tailed Z tests, power in testing of hypothesis, relation between normal and standard normal variable, and more. It also includes references for further reading.

Assignment 2
PSYC 2525

a. Two tailed Z test
i. H0: ( μ=μ0 )
ii. H1: ( μ≠μ0 )
b. One Tailed Z Test
i. H0: ( μ≤μ0 ) or ( μ≥μ0 )
ii. H1: ( μ> μ0 ) (right tail) or ( μ< μ0 ) (left tail)
Standard Normal Table Used to Answer 2a, 2b, and 2c (Thomopoulos, Nick, 2018, p. 45-55).
a) Required area = P( 0≤Z≤2 ) = 0.47725
b) Required area = P( Z ≤1)=P ( −∞<Z≤1 ) =0. 5+ P ( 0≤Z≤1 ) = 0.5 + 0.341345 =
0.841345 = 0.84
c) P ( −∞<Z <α ) =0. 95=0 .5+0. 45=0 . 5+P ( 0≤Z≤1 .645 ) =P ( −∞<Z ≤1. 645 ) , so required
Z-score = 1.645 below which 95% of the values will be positioned.
Decision you
make
Fail to reject the null
hypothesis (Accept) Reject the null hypothesis
True
nature of
the null
hypothesi
s
The null
hypothesis
is really true
Correct Conclusion Type –I error
The null
hypothesis
is really
false
Type –II error Correct Conclusion

ANS: Power in testing of hypothesis is the technique of selecting the correct choice. Hence, it is
the probability that is associated with rejection of a false null hypothesis. Power of a test is
calculated as (1 – probability of Type –II error) or simply 1−β , where β is the probability of
accepting or failing to reject a false null hypothesis (Park & Hun Myoung, 2015).
ANS: The relation between normal (X) and standard normal variable is known to be as
Z = X−μ
σ , where μ is the mean and σ is the standard deviation of X variable.
a. X =78 => Z =78−50
10 =2 .8
b. X = 92 => Z =92−50
10 =4 .2
c. X = 38 => Z = Z =38−50
10 =−1 .2
d. X = 44 => Z =44−50
10 =−0 . 6

ANS:
a. For two tailed test of significance,
Null Hypothesis: H0: ( μ=40 ) and
Alternate Hypothesis: HA: ( μ≠40 )
The considerations are taken accordingly at specified level of significance (D'Agostino,
RalphB, 2017).
b. SEM (Standard Error of the Mean) =
σ
√ n = 5
√ 100 =0 .5
c. For one tailed test of significance,
Null Hypothesis: ( μ≤40 ) and
Alternate Hypothesis: ( μ>40 ) at given level of significance (Sullivan & Michael, 2015)
References
D'Agostino, RalphB. Goodness-of-fit-techniques. Routledge, 2017.
Park, Hun Myoung. "Hypothesis testing and statistical power of a test." (2015).
Sullivan III, Michael. Fundamentals of statistics. Pearson, 2015.

Thomopoulos, Nick T. "Standard Normal." Probability Distributions. Springer, Cham, 2018. 45-55.