Z-Scores Z-scores

Z-Scores

Z-scores

Section1: Z-scores in SPSS

Z-scoresare type of descriptive statistics that presents the number of thestandard deviations between every data point and the mean. A sampleof 13 students was taken to act as a representative of their classand the Z-scores calculated using Excel. The results are presented inthe tables below.

Descriptive Statistics

N

Mean

Std. Deviation

quiz1

105

7.47

2.481

quiz2

105

7.98

1.623

quiz3

105

8.05

2.322

quiz4

105

7.80

2.280

quiz5

105

7.87

1.765

Final

105

61.48

7.943

Total

105

100.57

15.299

Percent

105

80.34

12.135

Valid N (listwise)

105

Zscore: Final

Zscore: Total

Zscore: Percent

Mean

Mean

Mean

ID

106484

-1.06707

-1.34458

-1.34672

108642

-.94118

-.29880

-.27547

127285

-.56351

-.16807

-.19306

132931

.82128

.15874

.13656

140219

.56950

.48554

.46617

142630

1.57663

1.40061

1.45502

153964

.19183

.74699

.79579

154441

1.19896

1.26988

1.29021

157147

1.57663

1.46597

1.45502

164605

1.70252

1.53133

1.53743

164842

-.31173

-.23343

-.19306

167664

1.19896

1.13916

1.12540

175325

.94717

.68163

.71338

Apositive z-score indicates that the data point considered is abovethe mean, while a negative z-score implies that the data point inquestion is below the mean.

Section2: Case Studies of Type I and Type II Errors

Thetype I Error result when the null hypothesis is rejected when it isin fact true. Formany disciplines, the classical hypothesis testing is critical inanalyzing the research data. In a typical condition, a small sampleof data is assumed to have an alpha significance level of .05.Normally, null hypothesis depicts lack of relationship between two ormultiple variables. To stimulate this, a situation can be assumed. Inthis case, considering an experimental drug supposed to reduce theblood pressure but suspected to induce cancer, the drug may be firstadministered to the rodents. When the tumor rate is known to be 10%among untreated animals, the null hypothesis can be “the rate oftumor in treated animals is less than or is equal to 10%”. In thiscase, the drug is safe and does not cause any tumors. On the otherhand, the alternative hypothesis may be “tumor rate in treatedanimals is over 10%”.

AType I error results when a true null hypothesis is rejected. In thecase provided, Type I error results when conclusions are made thatthe drug is unsafe when absolutely it is. On the other hand, a TypeII error results from failure of rejecting a false null hypothesis.In the case provided, Type II error results when it is concluded thatthe drug is safe when truly it is not.

Section3: Case Studies of Null Hypothesis Testing

Nullhypothesis refers to a statement that is considered to be truethroughout the entire analysis. A dubious judgment statistician wantsto determine whether the revolver with the capacity of holding sixcartridges is well loaded. Therefore, he proposes to run a nullhypothesis of the gun not being loaded through spinning of thecylinder in order to align one amongst the six chambers with barrelrandomly, then pulling the trigger two times consecutively. If gunwill go off when the trigger is pulled, then he will conclude thatwas loaded, hence reject the null hypothesis. On the other hand, ifthe gun will go off, he will be justified to conclude that the gunwas not loaded. Thereafter, the significance level of this test willbe determined.

Ifthe gun will not be loaded, it will not go off until the trigger ispulled, hence the test cannot erroneously reject the null hypothesis.This will result to a significance level of zero. Supposing only onechamber is loaded, the chance of the gun going off is 100 percentless the chance that it fails to go off.

Reference

Klein, J. (2014). Assessing University Students’ Achievements by Means of Standard Score (Z Score) and Its Effect on the Learning Climate. Studies in Educational Evaluation .