One-Sample Hypothesis Testing for Population Means and Proportions

  • Sampling Distribution: The probability distribution of a statistic obtained by selecting random samples from a population.

  • It describes how the sample means or proportions vary from sample to sample, providing the foundation for hypothesis testing. The central limit theorem plays a crucial role in this context, as it states that the sampling distribution of the sample mean will tend to be normally distributed as the sample size increases, regardless of the shape of the population distribution.


  • Sampling Distribution: The probability distribution of a statistic (such as the sample mean xˉxˉ or sample proportion p^p^​) obtained by selecting random samples of size nn from a population.

    • It describes how a sample statistic varies from sample to sample, serving as the theoretical foundation for statistical inference, confidence intervals, and hypothesis testing.

  • Central Limit Theorem (CLT):

    • States that for a random sample of size nn drawn from a population with mean μμ and variance σ2σ2, the sampling distribution of the sample mean xˉxˉ approaches a normal distribution N(μ,σ2n)N(μ,nσ2​) as sample size nn increases (n≥30n≥30), regardless of the shape of the underlying population distribution.

    • Standard Error of the Mean: σxˉ=σnσxˉ​=n​σ​.

  • Hypothesis Testing Framework:

    • Null Hypothesis (H0H0​): The default or status-quo claim about a population parameter, which always contains the equality condition (==, ≤≤, or ≥≥).

    • Alternative Hypothesis (H1H1​): The research claim or statement to be supported by sample evidence, formulated using >>, <<, or ≠=.

  • Errors in Hypothesis Testing:

    • Type I Error (Level of Significance, αα): The probability of rejecting H0H0​ when H0H0​ is actually true (α=P(Reject H0∣H0 is true)α=P(Reject H0​∣H0​ is true)).

    • Type II Error (ββ): The probability of failing to reject H0H0​ when H0H0​ is false (β=P(Fail to reject H0∣H0 is false)β=P(Fail to reject H0​∣H0​ is false)).

  • One-Sample Hypothesis Tests for Population Mean (μμ):

    • Known Population Variance (σ2σ2):

    • Test statistic: zstat=xˉ−μ0σn∼N(0,1)zstat​=n​σ​xˉ−μ0​​∼N(0,1).

    • Unknown Population Variance (σ2σ2):

    • Test statistic: tstat=xˉ−μ0sn∼t(n−1)tstat​=n​s​xˉ−μ0​​∼t(n−1), where ss is the sample standard deviation.

  • One-Sample Hypothesis Tests for Population Proportion (pp):

    • Applicable when np0≥5np0​≥5 and n(1−p0)≥5n(1−p0​)≥5.

    • Sample proportion: p^=xnp^​=nx​.

    • Test statistic under H0H0​: zstat=p^−p0p0(1−p0)n∼N(0,1)zstat​=np0​(1−p0​)​​p^​−p0​​∼N(0,1).

  • Decision Rules for Decision Making:

    • Critical Value Approach: Reject H0H0​ if the test statistic falls into the rejection region determined by αα

    • p-value Approach: Reject H0H0​ if p-value≤αp-value≤α.