﻿ One-Sample Z Test

# One-Sample Z Test

### Statistics > Basic Statistics > 1-Sample Z

Determines whether a sample from a normal distribution with known standard deviation could have a given mean.

Use One-Sample Z Test to compute a confidence interval or perform a hypothesis test of the mean when s is known. For a two-tailed one-sample Z

H0 : µ = µ0   versus   H1: µ ≠ µ0

where m is the population mean and µ0 is the hypothesized population mean.

### Dialog box items

Samples in columns:
Choose if you have entered raw data in columns.

oSamples: Enter the columns containing the sample data.

Summarized data:
Choose if you have summary values for the sample size, mean, and standard deviation.

oSample size: Enter the value for the sample size.

oSample Mean: Enter the value for the sample mean.

Report:
The display of outputs of VisualStat.

### Data

Data column must be numeric. You can generate a hypothesis test or confidence interval for more than one column at a time. Missing values are ignored.

### Charts

Displays a histogram, a histogram with a normal curve, and a boxplot.

Histogram:
Choose to display a histogram for each variable

Histogram with Normal Curve:
Choose to display a histogram with a normal curve for each variable

Boxplot of data:
Choose to display a boxplot for each variable

Options: Choose the options you want.

oExclude missing values: Check to excludes rows that have missing values.

oInverted Bar: Check to reverse the axes.

Options

Hypothesized mean:
Enter the test mean µ0

Standard deviation:
Enter the value for the population standard deviation.

Alternative Hypothesis:
Enter Two-sided, Upper One-sided, or Lower One-sided. If you choose an One-sided hypothesis test, an upper or lower confidence bound will be constructed, respectively, rather than a confidence interval.

Confidence Level:
Enter the level of confidence desired. Enter any number between 0 and 100. Entering 90 will result in a 90% confidence interval. The default is 95%.

### Example

This data set was collected by Bob Zarr of NIST in January, 1990 from a heat flow meter calibration and stability analysis. The response variable is a calibration factor.

The motivation for studying this data set is to illustrate a well-behaved process where the underlying assumptions hold and the process is in statistical control.

1.Open the DataBook compare.vstz

2.Select the sheet zarr13

3.Choose the tab Statistics, the group Basic Statistics and the command 1-Sample Z

4.Select group Samples in columns

5.In Sample, select Calib.

6.Click Options page. In Hypothesized mean, enter 9.265. In Standard deviation, enter 0.02.

7.Click OK

Report window output

One-Sample Z Test

Test of mu = 9.265 vs not = 9.265

The assumed standard deviation = 0.02

alternative hypothesis: true mean is not equal to 9.265

N    Mean   StDev  SE Mean  95% Conf Interval   z-Stat   Proba  Alt Hypothesis

Calib  195  9.2615  0.0228   0.0014   [9.2587; 9.2643]  -2.4711  0.0135          Accept

Interpreting the results

The test statistic, Z, for testing if the population mean equals 9.265 is -2.4711. The p-value, or the probability of rejecting the null hypothesis when it is true, is 0.0135. This is called the attained significance level, p-value, or attained α of the test. Because the p-value of 0.0135 is smaller than commonly chosen α-levels, there is significant evidence that μ is not equal to 9.265, so you can reject the null hypothesis in favor of μ not being 9.265.