This document provides an overview of single sample z-tests for inferential statistics. It discusses testing the population mean when the variance is known or unknown, and for small or large sample sizes. An example is given of testing if the average phosphate in blood is less than 4.8 mg/dl. The key steps of the z-test are outlined as defining hypotheses, calculating the test statistic, determining the p-value using z-tables or software, and rejecting the null hypothesis for low p-values. Formulas are presented for the z-test, t-test, chi-square test, and proportion z-test, along with considerations for each.

1. Introduction to Data Analytics
Lecture: Inferential Statistics – Single sample tests
NPTEL MOOC
By
Prof. Nandan Sudarsanam, DoMS, IIT-M and
Prof. B. Ravindran, CS&E, IIT-M

2. Single sample z-test
• What are you testing? Population Mean
• Known variance and unknown variance
• Small sample size and large sample size
• Example: Average phosphate in blood is less than 4.8 mg/dl, with a
known standard deviation of 0.4 mg/dl
• Data: 4.1, 3.9, 5.3, 4.7………

3. Single sample z-test
• Using the rubric for this example:
• Have a null and alternate hypothesis; H0:𝜇0 ≤ 4.8 and Halt:𝜇0 > 4.8
• Do some basic calculations/arithmetic on the data to create a single number
called the “test statistic”; 𝑧𝑠𝑡𝑎𝑡 =
𝑥−𝜇0
𝜎
𝑛
• If we assume the null hypothesis to be true (and make some assumptions
about the distributions of various variables), then the ‘test statistic’ should be
no different than a single random draw from a specific probability
distribution. This is the Z-distribution or N(0,12)
• Test the probability that the “test statistic” you calculated belongs to this
theoretical distribution. This is the p-value!; Use Z-tables, Excel, Matlab or R
• Low enough p-value is grounds for rejecting the null hypothesis

4. Single sample z-test
• Why is the z-stat a z distribution (or in other words a 𝑁 0, 12
)
𝑁 𝜇, 𝜎2
𝑁 𝜇, (𝜎
𝑛
)2
Now subtract 𝜇
𝑁 0, (𝜎
𝑛
)2
Now divide by 𝜎
𝑛
𝑁 0, (1)2

5. Single sample z-test
• The p-value is the probability of seeing a test statistic as
extreme as the calculated value if the null hypothesis is true.
• If 𝑧𝑠𝑡𝑎𝑡 was computed to be 1.2 then
• P-value, Based on the standard null hypothesis:
• H0: 𝜇 ≤ 4.8
• If null hypothesis was:
• H0: 𝜇 ≥ 4.8
• H0: 𝜇 = 4.8 (two tailed)
• Excel for right tail: =1-NORM.S.DIST(1.2,TRUE)
• Excel for left tail = NORM.S.DIST(1.2,TRUE)
• Excel for two-tail = 2*(1-NORM.S.DIST(1.2,TRUE))

6. Examples and formulas
Single Sample Tests What are you testing Example
z-test mean Phosphate in blood
t-test mean Phosphate in blood
Chi-Square test standard deviation Equal treatment
Proportion z-test proportion/likelihood Defective products
𝑡 =
𝑥−𝜇
(𝑠
𝑛
)
; df = n-1
z=
𝑥−𝜇
(𝜎
𝑛
)
χ2=(𝑛 − 1)
𝑠2
𝜎0
2; df = n-1
𝑧 =
𝑝 − 𝑝0
𝑝(1 − 𝑝0)
𝑛
• Unknown
Variance
• Small sample
size
• Using DOF
• Excel uses T.DIST,
T.DIST.RT, and
T.DIST.2T
• Binomial Approximation to
the Normal
• Also needs DOF
• Excel uses
CHISQ.DIST and
CHISQ.DIST.RT