Hypothesis Testing Procedure for a Population Mean with Known Standard Deviation
Overview
This section pertains to hypothesis testing for a population mean when the population standard deviation (Sigma) is known.
The section is labeled 8.2: "Hypothesis Test for a Population Mean Standard Deviation Known."
The known standard deviation must be explicitly provided to utilize this procedure, differentiating it from the method discussed in section 8.3, which applies when Sigma is unknown.
Assumptions for Hypothesis Testing
Required Sample Conditions
To correctly perform a hypothesis test for a population mean when Sigma is known, certain assumptions must be satisfied:
A simple random sample is needed. While the condition requires this, it's generally accepted that if valid information about a sample is given, it can be treated as legitimate.
At least one of the following must hold:
Large Sample Size: The sample size must be larger than 30 (i.e., $n > 30$).
Normal Population Distribution: If the sample size is smaller than 30, it is acceptable if the population is known to be normally distributed.
Hypotheses Setup
Null and Alternative Hypotheses
There are three ways to set up the hypotheses:
Null Hypothesis ($H0$): The population mean ($bc$) is equal to a specified value ($bc0$).
The value of $bc_0$ is provided in the testing information.
Alternative Hypotheses ($H_1$): The setup depends on the nature of the test:
Left-Tailed Test: $H1: bc < bc0$ (testing if the population mean is less than $bc_0$).
Right-Tailed Test: $H1: bc > bc0$ (testing if the population mean is greater than $bc_0$).
Two-Tailed Test: $H1: bc eq bc0$ (testing if the population mean is not equal to $bc_0$).
It is important to deduce the type of test based on how the hypothesis questions are worded, as these indicate what hypothesis setup is appropriate.
Test Statistic
Formula for Test Statistic
The test statistic for this hypothesis test is a z-score calculated using the following formula:
Z = \frac{\bar{X} - \mu_0}{\frac{\sigma}{\sqrt{n}}}Where:
$ar{X}$ = sample mean
$bc_0$ = population mean value under the null hypothesis
$sigma$ = population standard deviation
$n$ = sample size
The test statistic should typically be rounded to three decimal places unless otherwise specified.
Decision Making
Concluding the Hypothesis Test
The decision from a hypothesis test will fall into one of two categories:
Reject the Null Hypothesis ($H0$): Concludes that the data provides enough evidence to discard $H0$.
Do Not Reject the Null Hypothesis: Concludes that there is insufficient evidence to discard $H_0$.
Techniques for Decision Making
There are two methods for making decisions in hypothesis testing:
Critical Value Method:
Utilizes a table (specifically Table A3 from Chapter 7) to determine critical values.
This method is often easier for hand calculations since it requires looking up values in a table.
P-Value Method:
This method calculates the p-value which helps to determine how extreme the test statistic is under the null hypothesis.
More commonly used in technology-based hypothesis testing (e.g., with software such as Excel).
The primary focus will be on the critical value method, while the p-value method will be discussed briefly due to its reliance on technology for calculation.
Examples
Upcoming sections will illustrate examples where the hypothesis testing procedures outlined above will be implemented in practical scenarios.
In hypothesis testing, the "claim" refers to the specific statement or assertion about a population parameter that we intend to investigate. It's what the researcher or the problem is essentially trying to prove or disprove. The claim is inherently associated with a population parameter, such as the population mean (\mu), proportion (p), or standard deviation (\sigma). In the context of the provided note, which focuses on hypothesis testing for a population mean, the claim would specifically be about the value of the population mean (\mu).
To identify the claim when setting up a hypothesis test, you need to carefully read the problem statement and look for keywords that suggest what is being tested. The claim can be expressed in two forms:
The Null Hypothesis (H0): This is a statement of no effect or no difference, often stating that the population parameter is equal to a specified value ($\mu = \mu0 in our case). This is generally the easier one to write as it always contains an equality sign.
The Alternative Hypothesis (H_1): This is the statement that contradicts the null hypothesis, representing what we are trying to find evidence for. It can take one of three forms, depending on the nature of the claim:
Less than: \mu < \mu_0 (Left-Tailed Test)
Greater than: \mu > \mu_0 (Right-Tailed Test)
Not equal to: \mu \ne \mu_0 (Two-Tailed Test)
How to identify the claim:
Look for what is being suggested or questioned: If the problem explicitly states "we want to test if the mean is greater than X", then "\mu > X is your initial claim." If it says "there is no difference," then "\mu = X is the claim."
Identify the population parameter: In your scenario, it's the population mean (\mu).
Determine which hypothesis contains the original claim:
If the original claim contains an equality (e.g., "the mean is 50"), then the claim is the null hypothesis (H_0: \mu = 50).
If the original claim does not contain an equality (e.g., "the mean is less than 50" or "the mean is not equal to 50"), then the claim is the alternative hypothesis (H1: \mu < 50 or H1: \mu \ne 50).
By following these steps, you can accurately identify the claim and correctly set up your null and alternative hypotheses, which is crucial for the rest of the hypothesis testing procedure.
In hypothesis testing, the "claim" refers to the specific statement or assertion about a population parameter that we intend to investigate. It's what the researcher or the problem is essentially trying to prove or disprove. The claim is inherently associated with a population parameter, such as the population mean (\mu), proportion (p), or standard deviation (\sigma). In the context of the provided note, which focuses on hypothesis testing for a population mean, the claim would specifically be about the value of the population mean (\mu).
To identify the claim when setting up a hypothesis test, you need to carefully read the problem statement and look for keywords that suggest what is being tested. The claim can be expressed in two forms:
The Null Hypothesis (H0): This is a statement of no effect or no difference, often stating that the population parameter is equal to a specified value ($\mu = \mu0 in our case). This is generally the easier one to write as it always contains an equality sign.
The Alternative Hypothesis (H_1): This is the statement that contradicts the null hypothesis, representing what we are trying to find evidence for. It can take one of three forms, depending on the nature of the claim:
Less than: \mu < \mu_0 (Left-Tailed Test)
Greater than: \mu > \mu_0 (Right-Tailed Test)
Not equal to: \mu \ne \mu_0 (Two-Tailed Test)
It is important to deduce the type of test based on how the hypothesis questions are worded, as these indicate what hypothesis setup is appropriate.
How to identify the claim:
Look for what is being suggested or questioned: If the problem explicitly states "we want to test if the mean is greater than X", then "\mu > X is your initial claim." If it says "there is no difference," then "\mu = X is the claim."
Identify the population parameter: In your scenario, it's the population mean (\mu).
Determine which hypothesis contains the original claim:
If the original claim contains an equality (e.g., "the mean is 50"), then the claim is the null hypothesis (H_0: \mu = 50).
If the original claim does not contain an equality (e.g., "the mean is less than 50" or "the mean is not equal to 50"), then the claim is the alternative hypothesis (H1: \mu < 50 or H1: \mu \ne 50).
By following these steps, you can accurately identify the claim and correctly set up your null and alternative hypotheses, which is crucial for the rest of the hypothesis testing procedure.