Fundamentals of Hypothesis Testing: One-Sample Tests (Chapter 9 Study Notes)

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Fundamentals of Hypothesis Testing: One-Sample Tests

What is a Hypothesis?

A hypothesis is a claim or assertion about a population parameter, such as the population mean or population proportion. Hypothesis testing is a fundamental statistical method used to make inferences about populations based on sample data.

Population Mean (μ): The average value of a variable in the entire population. Example: The mean monthly cell phone bill in a city is μ = $42.
Population Proportion (π): The fraction of the population possessing a certain characteristic. Example: The proportion of adults in a city with cell phones is π = 0.88.

The Null Hypothesis (H0)

The null hypothesis (H0) states the claim or assertion to be tested. It is always about a population parameter, not a sample statistic.

Assume H0 is true at the start (similar to "innocent until proven guilty").
Represents the current belief or status quo.
Always contains "=", "≤", or "≥" signs. Example: The mean diameter of a manufactured bolt is 30mm (H0: μ = 30).

The Alternative Hypothesis (H1 or Ha)

The alternative hypothesis (H1) is the opposite of the null hypothesis and challenges the status quo.

Never contains "=", "≤", or "≥" signs.
May or may not be proven.
Generally, the hypothesis the researcher is trying to prove. Example: The mean diameter of a manufactured bolt is not equal to 30mm (H1: μ ≠ 30).

The Hypothesis Testing Process

Hypothesis testing involves several steps to determine whether to reject the null hypothesis based on sample data.

State the claim: For example, H0: μ = 50, H1: μ ≠ 50.
Sample the population: Collect sample data and calculate the sample mean (e.g., X̄ = 20).
Compare sample mean to claimed mean: If the sample mean is significantly different from the claimed mean, the null hypothesis may be rejected.
Decision: If the probability of obtaining such a sample mean is very small under H0, reject H0.

The Test Statistic and Critical Values

The test statistic measures how far the sample statistic is from the population parameter under H0. Critical values define the boundaries of the rejection region.

If the sample mean is close to the stated population mean, do not reject H0.
If the sample mean is far from the stated population mean, reject H0.
The critical value answers "how far is far enough" to reject H0.

Risks in Decision Making: Type I and Type II Errors

Hypothesis testing involves two types of errors:

Type I Error (α): Rejecting a true null hypothesis (false alarm). Probability is α (level of significance).
Type II Error (β): Failing to reject a false null hypothesis (missed opportunity). Probability is β.

Possible Hypothesis Test Outcomes

Decision	H0 True	H0 False
Do Not Reject H0	Correct Decision Confidence = 1 - α	Type II Error P(Type II Error) = β
Reject H0	Type I Error P(Type I Error) = α	Correct Decision Power = 1 - β

Additional info:

Confidence coefficient (1-α): Probability of not rejecting H0 when it is true.
Confidence level: (1-α) × 100%.
Power of a test (1-β): Probability of rejecting H0 when it is false.

Type I & II Error Relationship

Type I and Type II errors cannot happen at the same time.
Increasing α (Type I error probability) decreases β (Type II error probability), and vice versa.

Factors Affecting Type II Error (β)

β increases when the difference between the hypothesized parameter and its true value decreases.
β increases when α decreases.
β increases when population standard deviation (σ) increases.
β decreases when sample size (n) increases.

Level of Significance and the Rejection Region

The level of significance (α) determines the size of the rejection region. In a two-tail test, the rejection region is split between both tails of the sampling distribution.

Critical values mark the boundaries of the rejection region.
Example: H0: μ = 30, H1: μ ≠ 30, α = 0.05, critical values at ±1.96.

Hypothesis Tests for the Mean

There are two main types of hypothesis tests for the mean, depending on whether the population standard deviation (σ) is known or unknown.

σ Known: Use the Z test.
σ Unknown: Use the t test.

Z Test of Hypothesis for the Mean (σ Known)

When σ is known, convert the sample mean (X̄) to a Z test statistic:

The test statistic is:

Critical Value Approach to Testing

For a two-tail test for the mean (σ known):

Convert sample mean to ZSTAT.
Determine critical Z values for α from tables or software.
Decision Rule: If ZSTAT falls in the rejection region, reject H0; otherwise, do not reject H0.

Two-Tail Tests

Two critical values (±Zα/2) define the rejection regions in both tails.

Reject H0 if ZSTAT < -Zα/2 or ZSTAT > +Zα/2.

Steps in the Critical Value Approach to Hypothesis Testing

State the null and alternative hypotheses.
Choose the level of significance (α) and sample size (n).
Determine the appropriate test statistic and sampling distribution.
Determine the critical values dividing rejection and nonrejection regions.
Collect sample data, organize results, and compute the test statistic.
Make the statistical decision and state the managerial conclusion.

Hypothesis Testing Example (Critical Value Approach)

Claim: True mean diameter of a manufactured bolt is 30mm (σ = 0.8).
Hypotheses: H0: μ = 30, H1: μ ≠ 30 (two-tail test).
Significance level: α = 0.05, n = 100.
Critical values: ±1.96.
Sample results: X̄ = 29.84, σ = 0.8.
Test statistic:
Decision: Since -2.0 < -1.96, reject H0. There is sufficient evidence that the mean diameter is not equal to 30.

p-Value Approach to Testing

The p-value is the probability of obtaining a test statistic equal to or more extreme than the observed sample value, given H0 is true.

The p-value is also called the observed level of significance.
It is the smallest value of α for which H0 can be rejected.
Decision Rule: If p-value < α, reject H0; if p-value ≥ α, do not reject H0.

The 5 Step p-value Approach to Hypothesis Testing

State the null and alternative hypotheses.
Choose the level of significance (α) and sample size (n).
Determine the appropriate test statistic and sampling distribution.
Collect sample data, compute the test statistic and p-value.
Make the statistical decision and state the managerial conclusion.

p-Value Hypothesis Testing Example

Claim: True mean diameter of a manufactured bolt is 30mm (σ = 0.8).
Hypotheses: H0: μ = 30, H1: μ ≠ 30 (two-tail test).
Significance level: α = 0.05, n = 100.
Sample results: X̄ = 29.84, σ = 0.8.
Test statistic: ZSTAT = -2.0.
p-value calculation: For Z = -2.0, p-value = 0.0228 (left tail) + 0.0228 (right tail) = 0.0456.
Decision: Since p-value = 0.0456 < α = 0.05, reject H0. There is sufficient evidence that the mean diameter is not equal to 30.