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Unit 12 - Hypothesis Testing
Introduction to Statistics I (University of Calgary)
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Hypothesis Testing – 1
Unit 12: Hypothesis Testing
Textbook: 8.1 – 8.5
Objectives
Be able to understand and apply the steps of hypothesis testing to make
inferences about a population mean µ
Be able to interpret the components of a hypothesis test (hypotheses, test
statistic, p-value, conclusion, etc.) in the context of a specific scenario
Know when to calculate a z-score (zcalc) versus a t-score (tcalc) for a test statistic
value
Be able to understand the relationship between hypothesis testing and
confidence intervals
Motivation
The last few weeks of this class have focused heavily on the idea of statistical
inference: taking a sample from a larger population, analyzing the sample, and using
the results of this analysis to make a claim about the population from which the sample
was drawn.
Example
A researcher is interested in the average weight of adult brown bears. She takes a
sample of n = 10 bears and records their weights.
Population: all adult brown bears
Population parameter of interest: μ, the average weight of all adult brown bears
Sample: 10 bears
Sample statistic of interest: x̅, the average weight of the 10 bears in the sample
In Unit 11, we discussed confidence intervals, one method of statistical inference. A
confidence interval is constructed based on sample data and is an interval estimate of
the target population parameter. It gives us a range of values that are plausible values
for the population parameter.
Example
Based on the sample of 10 bears, a 90% confidence interval for μ is computed as
(387.89, 404.58). That is, there is 90% confidence that the average weight of all adult
brown bears is somewhere between 387.89 pounds and 404.58 pounds.
Downloaded by Kelvin Mulimi ()
, lOMoARcPSD|56450245
Hypothesis Testing – 2
As you can see, a confidence interval is useful for inference as it allows us to make a
claim about a population based on a sample drawn from that population. These
intervals can be especially useful if you might not have a specific value in mind for your
population parameter but rather just want to see a range of likely values.
In other situations, however, we might be interested in how the population parameter
relates to a specific value.
Examples
Is the average weight of adult male brown bears less than 400 pounds?
Is the average age of University of Calgary first-year students at least 19.8 years?
Does the average battery life of the new iPhone X differ from 11 hours?
Each of the above examples involve examining how a population mean (average
weight, average age, average time, respectively) relates to a specific value (400
pounds, 19.8 years, and 11 hours, respectively).
In this set of notes, we will discuss our second method of statistical inference,
hypothesis testing, which is appropriate for addressing questions similar to those in the
examples above, when we are interested in a particular value of a parameter.
Hypothesis Testing (for a Single Population Mean μ)
Before we begin, it is worth noting that hypothesis testing can be done for basically any
population parameter – a mean, a proportion, a median, a standard deviation, a
difference between means, etc. If you go on to STAT 217, you will see a wide variety of
hypothesis tests for different parameters. But for our purposes in this class, we will only
focus on hypothesis testing for a single population mean μ.
Conditions
When we’re interested in conducting a hypothesis test for a single population mean, we
can either carry out a 1-sample z-test or a 1-sample t-test. Which test we choose has to
do with whether we know the value of the population standard deviation, σ.
Both of these tests are parametric tests* that can be used to test an inference about a
population mean. In order to use either of these tests, the following two conditions must
be met:
1. The sample was collected using simple random sampling (SRS)
2. The sample was drawn from a normally distributed or an approximately normally
distributed population
*A parametric test is one that makes certain assumptions about the population from which a sample is
drawn – the main assumption being normality or approximate normality.
Downloaded by Kelvin Mulimi ()
Unit 12 - Hypothesis Testing
Introduction to Statistics I (University of Calgary)
Scan to open on Studocu
Studocu is not sponsored or endorsed by any college or university
Downloaded by Kelvin Mulimi ()
, lOMoARcPSD|56450245
Hypothesis Testing – 1
Unit 12: Hypothesis Testing
Textbook: 8.1 – 8.5
Objectives
Be able to understand and apply the steps of hypothesis testing to make
inferences about a population mean µ
Be able to interpret the components of a hypothesis test (hypotheses, test
statistic, p-value, conclusion, etc.) in the context of a specific scenario
Know when to calculate a z-score (zcalc) versus a t-score (tcalc) for a test statistic
value
Be able to understand the relationship between hypothesis testing and
confidence intervals
Motivation
The last few weeks of this class have focused heavily on the idea of statistical
inference: taking a sample from a larger population, analyzing the sample, and using
the results of this analysis to make a claim about the population from which the sample
was drawn.
Example
A researcher is interested in the average weight of adult brown bears. She takes a
sample of n = 10 bears and records their weights.
Population: all adult brown bears
Population parameter of interest: μ, the average weight of all adult brown bears
Sample: 10 bears
Sample statistic of interest: x̅, the average weight of the 10 bears in the sample
In Unit 11, we discussed confidence intervals, one method of statistical inference. A
confidence interval is constructed based on sample data and is an interval estimate of
the target population parameter. It gives us a range of values that are plausible values
for the population parameter.
Example
Based on the sample of 10 bears, a 90% confidence interval for μ is computed as
(387.89, 404.58). That is, there is 90% confidence that the average weight of all adult
brown bears is somewhere between 387.89 pounds and 404.58 pounds.
Downloaded by Kelvin Mulimi ()
, lOMoARcPSD|56450245
Hypothesis Testing – 2
As you can see, a confidence interval is useful for inference as it allows us to make a
claim about a population based on a sample drawn from that population. These
intervals can be especially useful if you might not have a specific value in mind for your
population parameter but rather just want to see a range of likely values.
In other situations, however, we might be interested in how the population parameter
relates to a specific value.
Examples
Is the average weight of adult male brown bears less than 400 pounds?
Is the average age of University of Calgary first-year students at least 19.8 years?
Does the average battery life of the new iPhone X differ from 11 hours?
Each of the above examples involve examining how a population mean (average
weight, average age, average time, respectively) relates to a specific value (400
pounds, 19.8 years, and 11 hours, respectively).
In this set of notes, we will discuss our second method of statistical inference,
hypothesis testing, which is appropriate for addressing questions similar to those in the
examples above, when we are interested in a particular value of a parameter.
Hypothesis Testing (for a Single Population Mean μ)
Before we begin, it is worth noting that hypothesis testing can be done for basically any
population parameter – a mean, a proportion, a median, a standard deviation, a
difference between means, etc. If you go on to STAT 217, you will see a wide variety of
hypothesis tests for different parameters. But for our purposes in this class, we will only
focus on hypothesis testing for a single population mean μ.
Conditions
When we’re interested in conducting a hypothesis test for a single population mean, we
can either carry out a 1-sample z-test or a 1-sample t-test. Which test we choose has to
do with whether we know the value of the population standard deviation, σ.
Both of these tests are parametric tests* that can be used to test an inference about a
population mean. In order to use either of these tests, the following two conditions must
be met:
1. The sample was collected using simple random sampling (SRS)
2. The sample was drawn from a normally distributed or an approximately normally
distributed population
*A parametric test is one that makes certain assumptions about the population from which a sample is
drawn – the main assumption being normality or approximate normality.
Downloaded by Kelvin Mulimi ()
