Critical Value and Confidence Intervals

Critical Value

  • Excel outputs critical values during hypothesis testing, labeled as "crit" or "Critical".

  • Critical value: Dividing point between the rejection and non-rejection regions.

  • Depends on:

    • Distribution (t, Z, etc.)

    • Degrees of freedom (for t)

    • Significance level \,\alpha

    • Direction of alternative hypothesis

Critical Value Notation

  • Z{\alpha/2} and t{\alpha/2} for two-tailed tests.

  • Z{\alpha} and t{\alpha} for one-tailed tests.

  • Excel provides positive critical values; user accounts for direction.

Obtaining Critical Values

  • Online calculators.

  • Excel:

    • norm.inv() for standard normal Z.

    • t.inv() for left/right-tailed t-tests.

    • t.inv.2t() for two-tailed t-tests.

  • Tables (e.g., Appendix B).

Z Critical Values in Excel

  • Use norm.inv().

  • Left-tailed: norm.inv(\alpha, 0, 1).

  • Right-tailed: norm.inv(1-\alpha, 0, 1).

  • Two-tailed: norm.inv(\alpha/2, 0, 1) (lower); upper is the positive version.

Finding t Critical Values

  • Historically via tables (Appendix B).

  • Excel: t.inv() and t.inv.2t() functions.

Excel Two-Tailed t Critical Values

  • Use t.inv.2t(\alpha, DF).

  • \,DF = n - 1

Excel One-Tailed t Critical Values

  • Use t.inv().

  • Left-tailed: t.inv(\alpha, DF).

  • Right-tailed: t.inv(1-\alpha, DF).

Critical Value Decision Making

  • Is the test statistic more extreme than the critical value?

    • Yes: Reject the null hypothesis.

    • No: Do not reject the null hypothesis.

  • Consistent with P-value approach.

Estimation and Confidence Intervals

  • Point Estimate: Single approximate value for a population parameter.

  • Interval Estimate: Range of values likely to contain the population parameter.

  • Confidence Interval (CI): Range of numbers from sample data likely to include the true population parameter; an empirical interval estimate.

Two Sided Intervals

  • Most CIs are two-sided: point estimate ± margin of error.

  • Margin of error: Half-width of a CI; measure of uncertainty.

  • Margin of error = critical value * standard error.

Math for the CI

  • Statistic from data is at the center of the sampling distribution.

  • Use the sampling distribution to create an interval likely to contain the true population parameter.

Using the CLT

  • CLT: Central Limit Theorem.

  • CLT Condition: The population must be normally distributed or the sample size must be at least 30.

Starting from the CLT

  • Assume population standard deviation \sigma known.

  • Formula: \,\mu = \bar{x} \pm Z * \frac{\sigma}{\sqrt{n}}

Selecting Z

  • Pick low and high side Z values to capture a large percentage (confidence level) of the sampling distribution.

  • Common confidence levels: 90%, 95%, 99% (align with \,\alpha levels 0.10, 0.05, 0.01).

Building a Confidence Interval

  • For a symmetric 95% CI, use Z = ±1.96.

  • Interval: \,\bar{x} \pm 1.96 * \frac{\sigma}{\sqrt{n}}

Working through Fall 2020 Class Heights

  • Example: N = 53, sample mean = 67.75 inches, assume \,\sigma = 4.5 inches.

  • 95% CI: 67.75 ± 1.96 * 4.5/\,\sqrt{53} = 66.539 to 68.961 inches.

Lower and Upper Bounds

  • \,\bar{x} - Z{\alpha/2} * \frac{\sigma}{\sqrt{n}} < \mu < \bar{x} + Z{\alpha/2} * \frac{\sigma}{\sqrt{n}}

Standard Error and Critical Value

  • Margin of error Z_{\alpha/2} * \frac{\sigma}{\sqrt{n}}. \,\frac{\sigma}{\sqrt{n}}

  • is standard error.

  • Z_{\alpha/2} is a critical value.

About Levels and alpha

  • CI abbreviated, like 95% CI.

  • Higher confidence means larger Z and wider CI.

Unknown Population Standard Deviation

  • Substitute sample standard deviation s for \sigma.

  • Use Student’s t distribution with (n – 1) degrees of freedom.

  • \bar{x} \pm t_{\alpha/2} * \frac{s}{\sqrt{n}}

Recap of Computational Terms

  • Confidence Interval = Point Estimate ± Margin of Error.

  • Margin of Error = Critical Value * Standard Error

  • Confidence Level = 1 – \,\alpha

  • \,\bar{x} \pm Z_{\alpha/2} * \frac{\sigma}{\sqrt{n}},

  • if population \sigma known

  • \,\bar{x} \pm t_{\alpha/2} * \frac{s}{\sqrt{n}}, if population \sigma not known with (n – 1) DF

Converting Between The Forms

  • Lower Bound = Point Estimate – Margin of Error

  • Upper Bound = Point Estimate + Margin of Error

  • Point Estimate = (Lower Bound + Upper Bound) / 2

  • Margin of Error = Upper Bound – Point Estimate

Assigning Probability

  • CI for the mean is sample mean ± margin of error.

Normal Approximation Interval

  • Conditions for CI for Proportion

    • Sample size n is fixed and known.

    • There are exactly two options for each participant, occurrence or non-occurrence.

    • Each participant’s response is independent of all other participant’s responses.

    • The sample contains at least 5 occurrences and 5 non-occurrences.

Using Sample Proportion

  • Sample proportion \,\hat{p} = X/n.

  • Standard error \,\sqrt{\frac{\hat{p}*(1-\hat{p})}{n}}.

  • Critical value uses standard normal Z.

  • \,\hat{p} \pm Z_{\alpha/2} * \sqrt{\frac{\hat{p}*(1-\hat{p})}{n}}

Support or Reject?

  • Reject claimed values outside confidence interval.

  • Support claimed values inside confidence interval.

Claims are Hypotheses

  • H0: p = value using p for population proportion

  • HA: p ≠ value

  • Reject H0 if hypothesis value not inside confidence interval. Do not reject (support) H0 if hypothesis value is inside confidence interval.