# University of Washington Hypothesis Testing Questions

You are testing the tension strength of steel reinforcing bars you are making. You want to know if the minimum yield strength of your reinforcing bars is greater than 67,000 psi. You run 20 tests. You set a null hypothesis (H0) of strength = 67,000 psi and an alternative hypothesis (H1) that strength > 67,000 psi. You run a one-tailed t-test (since you are concerned with strengths GREATER than 67,000 psi only). Results indicate a p-value of 0.03 for the null hypothesis (H0). What does this p-value actually mean?

Want to find out from others what they say it means?

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Listen to this podcast: Planet Money – Episode 677: The Experiment ExperimentLinks to an external site.

https://www.npr.org/sections/money/2016/01/15/4632…

All that (and the course material) should arm you with enough information to say what a reported p-value actually means.

Advice: you do not need any statistical calculations to answer this question. I have given you all you need. Rather, I want you to concentrate on what the p-value actually means. I ask this because so many are incorrect on its meaning. No worries – the ASA’s statement and the Planet Money episode try and drive home its true meaning.

Lecture notes:

* Logic of Hypothesis Testing (p. 369-392) – Section XI.1 through XI.11. If you want to break it up, do this for week 4.

* Course Notes: Hypothesis Testing.pdf. Another title for this could be “Using statistics to examine the difference in means.” Traditionally, the way this is set up is with two hypotheses: the null and alternative.

HYPOTHESIS TESTING
1 INTRODUCTION
Hypothesis testing is a way in which statistical methods can be used to help in the decision making
process. Such testing considers the mean and standard deviation of a group of data, the
confidence level (a probability statement) and something about the population being sampled.
Hypothesis testing is extremely helpful in performing multiple regression analysis and hence it is
important for you to understand the basics.
Note: the methods described here assume that the data being considered are normally distributed.
While there is no certainty, it is often true that construction data are normally distributed (for
example, soil density, concrete strength, pipe tolerances). Often, we make the normal assumption
without checking. It is always good to check.
2 HYPOTHESES
Webster’s Seventh New Collegiate Dictionary defines hypothesis as “…a tentative assumption
made in order to draw out and test its logical or empirical consequences…an assumption or
concession made for the sake of argument…” You can begin to see the problem in explaining
hypothesis testing.
There are always two hypotheses for any statistical test (Blank 1980). These hypotheses are
H  null hypothesis (most important)
0
H  alternative hypothesis
1
What is about to be presented is one of the fundamental problems in statistics which is the use of
“double negative” statements. Any hypothesis must be tested statistically to be rejected or not
rejected. “Not rejected” is a statistical way of accepting something. Think of it as the equivalent of
“I can’t say no” rather than simply “yes.” The first statement is slightly less committed.
The hypotheses ( H or H ) can result in two types of errors if the wrong one is selected, as shown
0
1
in Table 1. The probability of the Type I and II errors is very important, since it determines how
carefully you must distinguish between true and false hypotheses.
(This is an area in which statistical “games” can be played, so you need to be very careful.) These
probabilities are (after Blank 1980).
Probability of a Type I error = α
1
Probability of a Type II error = β
Table 1. Types of Hypothesis Errors
“The Actual Decision”
“The Truth”
Reject H
Accept H
0
0
H true
Type I Error
Correct!
0
H
0 false
Correct!
Type II Error
The general form for calculating the z-statistic for hypothesis testing is
𝑧𝑐𝑎𝑙𝑐 =
where
sample mean
(𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛) − (ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑧𝑒𝑑 𝑣𝑎𝑙𝑢𝑒)
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟
=
x
hypothesized
value
=
μ (sometimes assumed to = 0 in regression
hypothesis testing)
standard error
=
σ
sample size
=
 standard deviation of means of
n
random samples of size n from a “parent”
population with standard deviation σ .
Standard error is sometimes designated
σ .
x
n
The same general form applies to the t-statistic for hypothesis testing when none of
the population statistics ( μ, σ ) are known:
𝑡𝑐𝑎𝑙𝑐 =
where
sample mean
(𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛) − (ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑧𝑒𝑑 𝑣𝑎𝑙𝑢𝑒)
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟
=
x
hypothesized
value
=
μ (again, sometimes an assumed or stated
value)
standard error
=
s
n
s
2
x
2.1 EXAMPLE 1: PCC MIX
For this example, use the data originally shown in the Data Distributions notes (Figure 1).
Figure 1. Histogram and the normal distribution for Portland cement concrete 28-day
compressive strength data (Willenbrock 1976).
This contractor states that the batch plant has produced a mix in the past of
μ  33.26 MPa
σ  2.67 MPa
(Since these are population statistics, you can assume that these data were collected over a long
period of time.)
On the job you take six samples (cylinders) with the result that x = 31.36 MPa.
Question: Is the contractor correct?
Solution: Assume that the data are normally distributed and use hypothesis testing.
3
H : μ  33.26 MPa
0
H : μ  33.26 MPa
1
z
calc

x μ
σ
n

31.36  33.26
2.67
6
 1.74
 1.65 (for Type I error (or α)  5%)
critical
(to get this value, use the z-statistic table at end of these notes. First, this is a one-tailed ztest since all we want to know is if the sampled values indicate a mean of 33.26. Therefore,
since the table is for a one-tailed test, go into the table and find 0.0500, or closest to it for a
negative z-statistic. This ends up being 0.0495 for a z-statistic of -1.65).
z
Where:
3.
μ
=
population mean for a specific test and tester group
Fixed value =
76 mm. for maximum specified slump ( H1 >76 mm.)
=
5% for air content (assumed based on WSDOT Spec.
6-02.3(2)A for cast-in-place concrete above the finished
ground line) H1  5%
=
27.6 MPa for 28-day compressive strength
t-statistic
t
x -μ 0
s
n
ν  n 1
where:
x
μ
0
s
n
s
n
=
sample mean for a specific test or tester groups
=
stated mean population value in H
=
=
standard deviation of the sample
sample size
=
standard error (sometimes designated s )
x
0
References:
Blank, Leland, Statistical Procedures for Engineering, Management, and Science, McGraw-Hill,
1980, p. 377.
Steel, Robert and Torrie, James, Principles and Procedures of Statistics, McGraw-Hill, 1960,
p. 19.
18
Figure 2. Illustrated results for Table 5Table 4.
19
Table 5. Results of Hypothesis Testing for Fixed Population Values
t-statistic
Critical Range (a)
Test
Air Content
(Basic)
Notes
(a)
(b)
Comparison
Overall = 5%
Calculated
+0.389
WSDOT = 5%
Industry = 5%
α  0.05
+2.120
Conclusion (b)
No significant difference
+1.018
+2.365
No significant difference
-0.161
-2.306
No significant difference
Critical region defined by the t-statistic for a two tail Type I error of 5% (α  0.05) for v  n  1 degrees of
freedom.
Conclusion based on hypothesis test described in Table 3.6
t-statistic
Critical Range (c)
Test
Slump
(Basic)
Compressive
Strength (d)
Comparison
Overall = 76 mm.
Calculated
+11.604
WSDOT = 76 mm.
α  0.05
> +1.746
Conclusion
Significant difference
+7.025
> +1.895
Significant difference
Industry = 76 mm.
+9.006
> +1.860
Significant difference
Overall = 27.6 MPa
+14.796
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