### 1 Sample Z test

## 1 Sample Z test

1 Sample Z test

Just like 1 Sample T test, 1 Sample Z test is also used to calculate the mean of a population or make comparison with a standard or a target value. It is also useful for baselining of the process as it gives a Confidence Interval of Mean with an upper & lower value, which means that Mean of Population will also lie within this range, with 95% probability. But an important input without which 1 sample Z does not work is the Standard Deviation. It asks for a Standard Deviation and terms it as “The Assumed Standard Deviation”.

Minitab navigation -> Stat – Basic Stats – 1 sample Z

z1

Select a Continuous value(Y) from the available fields and press “Ok”

z2

But here, we have to give a Standard Deviation value. Without this, the 1 Sample Z test would not work.

The following result appears in session window:

One-Sample Z: Production (Kg)

The assumed standard deviation = 40

Variable N Mean StDev SE Mean 95% CI

Production(Kg) 209 271.95 37.63 2.77 (266.52, 277.37)

Variable – Factor being measured

N –No of observations

Mean – Mean of sample

StDev – Standard Deviation of sample

SE Mean – Standard Error of Mean

95% CI – Confidence Interval of Population, with 95% probability. Also used for baselining of process.

Comparison with standard

We can also compare our data against a standard or target by selecting the “Perform Hypothesis Test” and giving a “Hypothesis Mean “ value, see image above

The following result appears in session window:

One-Sample Z

Test of mu = 50 vs not = 50

The assumed standard deviation = 40

N Mean SE Mean 95% CI Z P

200 250.00 2.83 (244.46, 255.54) 70.71 0.000

Z – Helps in determination of P-value for Z-test

P – P-value < 0.05 Alternate is true .i.e., Mean is not equal to target or standard

P-value ≤ 0.05 Null is true, i.e., Mean is equal to standard or target

Another way to understand this is as follows:

• If Population Mean is equal to Hypothesis Mean , then Null is True

• If Population Mean is not equal to Hypothesis Mean, then Alternative is True

• If Population Mean is less than Hypothesis Mean, then Alternative is True

• If Population Mean is Greater than Hypothesis Mean, then Alternative is True

Other Options

a. There are several other options in Minitab for data analysis through 1 Sample T. We can give specific values also such as Mean, sample count and SD

b. As mentioned earlier, we can change the P-value also in Minitab present in the “Options”. It also has 3 options for Alternate:

i) Less Than: Gives the upper limit of the CI

ii) Not Equal: Default option. Gives the CI

iii) Greater Than: Gives the lower limit of the CI

These options can also be used by changing the 95% probability value of the CI and by comparing it against a target value

Graphical Representation

The output numerical data can also be represented in graphical manner. Minitab gives 3 options – we can use either one of them and even all of them also:

• Histogram

• Individual value plot

• Boxplot

1 sample Z test is seldom used in real life as most of the times, standard deviation of the population is not known. Still it is a very easy yet powerful test and hence must be used if all the inputs are available.

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