Notes 9.1, 9.4 - Part I
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MAT 1450 – Introduction to Statistics
OS Chapter 9.1 – Null and Alternative Hypothesis
OS Chapter 9.4 – Rare Events, the Sample, Decision and Conclusion
MOM #17
Confidence intervals is one way to make statistical inferences about a population based upon a sample. Another way to do this is through a process called hypothesis testing
.
Ex. Previously, an organization reported that adults in a certain city consumed an average of 3 cups of coffee per day. The organization believes that the current average coffee consumption is lower. A random sample of 25 adults from the city was surveyed about their daily coffee intake. The sample mean was found to be 2.8 cups, with a sample standard deviation of 1.2 cups. Is this data enough evidence to say that the organization is wrong?
Hypothesis testing
is used to assess the plausibility of a hypothesis using sample data.
Developed by Neyman & Pearson in 1933 as a framework for applied decision making.
The purpose is to aid researchers in reaching a decision about a population by examining a sample.
Hypothesis testing is a statistical analysis of an assumption about a population parameter, typically the mean, standard deviation, or proportion.
Hypothesis testing begins with the “status quo” (
H
o
) and then compares it to an alternative situation (
H
a
∨
H
1
).
H
o
is always equal (
H
o
=
¿
¿
).
Ex. The average IQ for the adult population is 100 with a standard deviation of 15. A researcher believes this value has changed in recent years. The researcher decides to assess the IQ of 75 random adults.
H
0
=
100
H
a
≠
100
Ex. The NFL leadership currently believes the chance of the higher ranked team winning a coin toss is greater than 50%. The study a random sample of 200 coin tosses in NFL games.
H
0
=
0.5
H
a
>
0.5
1
Null Hypothesis (H
0
)
Alternative Hypothesis (H
1
)
Symbol
Symbol
Clue Words
Type of Test
=
<
Less than
Decreased
Left tailed test
=
>
More than
Increased
Right tailed test
=
≠
Not equal to
Has changed
Are different
Two tailed test
There are two methods traditionally used
o
Critical value 2
A p
value is a statistical measure (a probability) to determine the likelihood that an observed outcome is the result of chance.
Calculator
Stat
Tests
o
Z-Test (#1) if α
known
o
T-Test (#2) if s
known
o
p
-value
We will be using the p
-value method
Hypothesis Testing (p value approach)
Step 1
Determine hypothesis & Sketch
Null hypothesis will always be μ
=
¿
Alternative hypothesis is what is being tested
Step 2
Find significance level
If none is stated, use α
=
0.05
Step 3
Find test statistic
using appropriate formula
z
0
when σ
is known
t
0
when σ
is unknown
Step 4
Find the p value
using your calculator
Step 5
Compare
p value and
significance level to decide
If p
<
α
, reject the null hypothesis
If p
>
α
, fail to reject the null hypothesis
Step 6
Interpret
results
Write a sentence to answer the question
z
0
=
x
−
μ
0
σ
0
Ex. Previously, an organization reported that adults in a certain city
consumed an average of 3 cups of coffee per day. The organization
believes that the current average coffee consumption is lower. A
3
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random sample of 25 adults from the city was surveyed about their daily coffee intake. The sample mean was found to be 2.8 cups, with a sample standard deviation of 1.2 cups. Conduct a hypothesis test at a significance level of α
=
0.001
to test the claim.
Ex. It is commonly believed that the mean IQ score of the general population is 100. However, you are not entirely convinced and believe that the mean IQ score is higher than 100. You collect data from a sample of 36 individuals and find that their mean IQ score is 105 with a standard deviation of 12. Use a 0.05 significance level to test the claim.
4
Ex. A manufacturer is considering investing in a new machine for producing widgets. His current
machine produces and average of 25 widgets per second. He wants to know if this new machine produces a different number of widgets. A random sample of widget production rates from the new machine is collected, and the data points are as follows:
25.6, 26.2, 22.5, 20.5, 26.4, 27.4, 23.6, 26.9, 25.7, 24.9.
5
Assuming that the population follows a normal distribution, conduct a hypothesis test at the 0.01 significance level, and draw a conclusion.
6
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