1-Sample Z

 

Overview  |  How to  |  Data  |  Example

 

 

 

Overview

 

The 1-Sample Z conducts a hypothesis test of the mean based on sampled data when the population standard deviation, s is known.  It also calculates a confidence interval of the mean.

 

This procedure is based on normal distribution.  For larger samples (>30), the sample standard deviation may be used in the absence of the population standard deviation.  For smaller samples, make sure the data is drawn from a normal distribution or the t-test may be more appropriate.

 

Test

Null Hypothesis, H0

Alternate Hypothesis, H1

One-tailed

m = m0

m < m0 or m > m0

Two-tailed

m = m0

m  ¹ m0

 

where m is the population mean and m0 is the hypothesised population mean.

 

 

 

 

 

How to

 

1.       Choose ProcessMA > Basic Statistics > 1-Sample Z.

 

2.       In Variable, select the column containing the sample.

 

3.       In Sigma, enter the population standard deviation.

 

4.       In Test mean, enter the hypothesised population mean.

 

5.       Click OK.

 

 

Optional

 

6.       In Alternate, select the type of test.

 

7.       In Confidence level, enter the desired confidence level for the confidence interval.

 

8.       Check Plot dotplot, if you want to display a Dotplot of the data and the hypothesis test parameters.

 

 

Note       To select a column of data into a textbox, double-click on any of the column names shown in the list on the left of the dialog box while in the textbox.

 

 

 

 

 

Data

 

Variable: Numeric.

               

Sigma: Numeric.

 

Test mean: Numeric.

 

Confidence level: Numeric; Between 0 and 100.

 

 

 

 

 

Example

 

You measured the thickness of 30 new widgets made.  You know that the thicknesses of these parts are normally distributed with a standard deviation of 1.4.  You want to test if the population mean is 57 and also calculate a 90% confidence interval for the mean.

 

1.       Open worksheet ProcessMA > Tools > Data Files > Stat.xls.

 

2.       Choose ProcessMA > Basic Statistics > 1-Sample Z.

 

3.       In Variable, select A – Thickness.

 

4.       In Sigma, enter 1.4.

 

5.       In Test mean, enter 57.

 

6.       In Alternate, select Not equals.

 

7.       In Confidence level, enter 90.

 

8.       Check Plot dotplot.

 

9.       Click OK.

 

 

 

 

Interpretation

 

For a desired a = 0.05, since p = 0.0002 < a, we will reject H0.  Therefore, we conclude that there is significant evidence that the population mean is not equal to 57.

 

We can also observe from the Dotplot that the hypothesised value falls outside the 90% confidence interval for population mean (55.627, 56.468).  Therefore, H0 will also be reject for a = 0.10.