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Department of Statistics, KAU, 2019
© Textbook: Bluman, Allan G. (2012). Elementary Statistics : A Step by Step Approach, 8th Edition, McGraw-Hill.
Chapter 8: Hypothesis Testing
Introduction
Researchers are interested in answering many types
of questions.
Such questions can be answered through statistical
hypothesis testing, which is a decision-making
process for evaluating claims about a population.
8 – 1: Steps in Hypothesis Testing
A statistical hypothesis is an conjecture about a population
parameter. This conjecture may or may not be true.
The null hypothesis (𝑯𝟎) is a statistical hypothesis that states
that there is no difference between a parameter and a
specific value.
The alternative hypothesis (𝑯𝟏) is a statistical hypothesis
that states that there is a difference between a parameter and
a specific value.
8 – 1: Steps in Hypothesis Testing (cont.)
In this course, we are interested in statistical hypotheses that
compare the population mean (i.e., 𝜇) to a specified number
(e.g., k).
Types of tests: The null and alternative hypotheses are stated
together as follows:
Two-tailed
test
Right-tailed
test
Left-tailed test
Null
Hypothesis
𝑯𝟎: 𝝁 = 𝒌 𝑯𝟎: 𝝁 ≤ 𝒌 𝑯𝟎: 𝝁 ≥ 𝒌
Alternative
Hypothesis
𝑯𝟏: 𝝁 ≠ 𝒌 𝑯𝟏: 𝝁 > 𝒌 𝑯𝟏: 𝝁 < 𝒌
Example 8 – 1
State the null and alternative hypotheses for each
conjecture:
a) A researcher thinks that if expectant mothers use
vitamin pills, the birth weight of the babies will
increase. The average birth weight of the
population is 8.6 pounds.
Example 8 – 1 (cont.)
State the null and alternative hypotheses for each
conjecture:
a) A researcher thinks that if expectant mothers use
vitamin pills, the birth weight of the babies will
increase (>). The average birth weight of the
population is 8.6 (=k) pounds.
𝑯𝟎: 𝝁 ≤ 𝟖. 𝟔 and 𝑯𝟏: 𝝁 > 𝟖. 𝟔
Example 8 – 1 (cont.)
b) An engineer hypothesizes that the mean number of
defects can be decreased in a manufacturing
process of compact disks by using robots instead
of humans for certain tasks. The mean number of
defective disks per box is 18.
Example 8 – 1 (cont.)
State the null and alternative hypotheses for each conjecture:
b) An engineer hypothesizes that the mean number of defects can be decreased (<) in a manufacturing process of compact disks by using robots instead of humans for certain tasks. The mean number of defective disks per box is 18 (=k).
𝑯𝟎: 𝝁 ≥ 𝟏𝟖 and 𝑯𝟏: 𝝁 < 𝟏𝟖
Example 8 – 1 (cont.)
c) A teacher feels that a new teaching strategy will
change the scores of the students. The teacher is
not sure whether the scores will be higher or lower.
In the past, the mean of the scores was 73.
Example 8 – 1 (cont.)
c) A teacher feels that a new teaching strategy will
change the scores of the students. The teacher is
not sure whether the scores will be higher or
lower (≠). In the past, the mean of the scores was
73 (=k).
𝑯𝟎: 𝝁 = 𝟕𝟑 and 𝑯𝟏: 𝝁 ≠ 𝟕𝟑
8 – 1: Steps in Hypothesis Testing (cont.)
A statistical test uses the data obtained from a
sample to decide about whether the null hypothesis
should be rejected.
The numerical value obtained from a statistical test is
called the test value.
8 – 1: Steps in Hypothesis Testing (cont.)
A correct decision occurs if you reject the null
hypothesis when it is false or do not reject the null
hypothesis when it is true.
A type I error occurs if you reject the null hypothesis
when it is true.
A type II error occurs if you do not reject the null
hypothesis when it is false.
8 – 1: Steps in Hypothesis Testing (cont.)
The level of significance is the maximum probability of
committing a type I error. This probability is symbolized by 𝜶.
The p-value is the probability of getting a sample statistic
(such as the sample mean) in the direction of the alternative
hypothesis when the null hypothesis is true.
If the p-value is more than 𝜶, the decision is to not reject the
null hypothesis
If the p-value is less than 𝜶, the decision is to reject the null
hypothesis
8 – 2: Z–Test for a Mean
The Z-test is a statistical test for the mean of a
population. It can be used when the population is
normally distributed, and the population standard
deviation (i.e., 𝜎) is known. The formula for the Z-
test is
𝒁 =ഥ𝒙 − 𝒌
𝝈/ 𝒏
where ഥ𝒙 is the sample mean, 𝒏 is the sample size,
and 𝒌 hypothesized population mean.
8 – 2: Z–Test for a Mean (cont.)
A researcher wishes to see if the mean number of days that a basic, low-price, small automobile sits on a dealer’s lot is 29. A sample of 30 automobile dealers has a mean of 30.1 days for basic, low-price, small automobiles. The standard deviation of the population is 3.8 days. At a 0.05, test the claim that the mean time is:
a. greater than 29 days.
b. not equal to 29 days.
c. Less than 29 days.
Example 8 – 3
A researcher wishes to see if the mean number of days that a basic, low-price, small automobile sits on a dealer’s lot is 29 (=k). A sample of 30 (=𝒏) automobile dealers has a mean of 30.1 days (= ഥ𝒙)for basic, low-price, small automobiles. The standard deviation of the population is 3.8 (= 𝜎)days. At a 0.05, test the claim that the mean time is:
a. greater than 29 days. 𝑯𝟎: 𝝁 ≤ 𝟐𝟗 and 𝑯𝟏: 𝝁 > 𝟐𝟗
b. not equal to 29 days. 𝑯𝟎: 𝝁 = 𝟐𝟗 and 𝑯𝟏: 𝝁 ≠ 𝟐𝟗
c. Less than 29 days. 𝑯𝟎: 𝝁 ≥ 𝟐𝟗 and 𝑯𝟏: 𝝁 < 𝟐𝟗
Example 8 – 3 (cont.)
Write the following in Excel:
number of days (OR you can write anything here!)
30.1 (=sample mean)
3.8 (=population standard deviation)
30 (=sample size)
DO NOT CHANGE THE ORDER!
Example 8 – 3 (cont.)
Example 8 – 3 (a)
2. Input range!
1. Choose “summary input”
3. Input the value of k4. Select “greater than”
5. Change this to z-test!
Example 8 – 3 (a)
Example 8 – 3 (b)
2. Input range!
1. Choose “summary input”
3. Input the value of k
4. Select “not equal”
5. Change this to z-test!
Example 8 – 3 (b)
Example 8 – 3 (c)
2. Input range!
1. Choose “summary input”
3. Input the value of k4. Select “less than”
5. Change this to z-test!
Example 8 – 3 (c)
