Explain in detail the difference in the confidence interval estimate and the hypothesistesting procedure when you know and do not know the population standard deviation

Explain in detail the difference in the confidence interval estimate and the hypothesistesting procedure when you know and do not know the population standard deviation. What arethe similarities and differences?

The correct answer and explanation is:

Certainly! Here’s a detailed explanation comparing confidence interval estimation and hypothesis testing when the population standard deviation (σ) is known versus when it is unknown.

Confidence Interval and Hypothesis Testing: Known vs Unknown Population Standard Deviation

1. When Population Standard Deviation (σ) is Known:

• Confidence Interval Estimate:
  • The population standard deviation is known.
  • Use the Z-distribution (standard normal distribution).
  • The confidence interval (CI) for the population mean μ\mu is calculated as:
  xˉ±Zα/2×σn\bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} where xˉ\bar{x} is the sample mean, Zα/2Z_{\alpha/2} is the critical Z-value for the confidence level, σ\sigma is the known population standard deviation, and nn is the sample size.
• Hypothesis Testing:
  • Test statistic used is the Z-test:
  Z=xˉ−μ0σ/nZ = \frac{\bar{x} – \mu_0}{\sigma / \sqrt{n}} where μ0\mu_0 is the hypothesized population mean.
  • Since σ\sigma is known, the sampling distribution of the sample mean is normal, allowing the use of the Z-distribution.

2. When Population Standard Deviation (σ) is Unknown:

• Confidence Interval Estimate:
  • The population standard deviation is unknown.
  • Replace σ\sigma with the sample standard deviation ss.
  • Use the t-distribution with n−1n-1 degrees of freedom to account for additional uncertainty.
  • The confidence interval is:
  xˉ±tα/2,n−1×sn\bar{x} \pm t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}}
• Hypothesis Testing:
  • Use the t-test statistic:
  t=xˉ−μ0s/nt = \frac{\bar{x} – \mu_0}{s / \sqrt{n}}
  • The t-distribution is wider (has heavier tails) than the Z-distribution, reflecting greater uncertainty because σ\sigma is estimated from the sample.

Similarities:

• Both procedures (CI and hypothesis testing) aim to make inferences about the population mean.
• Both use sample data to estimate parameters.
• The structure of the test statistic is similar: (estimate – hypothesized value) divided by a measure of standard error.
• Both use critical values from a probability distribution (Z or t) based on the confidence level or significance level.
• When sample size nn is large, the t-distribution approaches the Z-distribution.

Differences:

• The key difference lies in whether σ\sigma is known:
  • Known σ\sigma: Use Z-distribution (more precise).
  • Unknown σ\sigma: Use t-distribution (accounts for added variability from estimating σ\sigma).
• The t-distribution depends on degrees of freedom, affecting the critical values.
• For small samples, using t-distribution is essential for accuracy when σ\sigma is unknown.
• Hypothesis testing decision rules differ slightly because critical values differ between Z and t distributions.

Summary:

Knowing the population standard deviation allows for straightforward use of the Z-distribution, simplifying calculations and resulting in narrower confidence intervals and more powerful hypothesis tests. When σ\sigma is unknown, the sample standard deviation ss replaces σ\sigma, and the t-distribution is used to account for the additional uncertainty, especially critical for small sample sizes. Both methods share fundamental goals and structures but differ in the precision and distribution used for inference.