Independent normal observations have known sigma. A one-sided test compares H0: mu = mu0 with H1: mu > mu0. Keep alpha and the true effect mu - mu0 > 0 fixed. Which change raises power?

• Reduce independent sample size while keeping variability unchanged
• Increase independent sample size with variability unchanged
• Increase variability while keeping independent sample size unchanged

What does planned power describe?

• Rejection probability at a specified true effect
• The probability that H0 is false after a significant result
• The probability that an estimated effect exceeds the useful-effect threshold

For the same fixed test and assumptions, power calculated from the observed effect supplies independent evidence beyond the test result.

An unrounded sample-size expression is E = (C / delta)^2, with C > 0 fixed. E = 144 at delta = 3. When delta is halved to 1.5, what is E before rounding?

• 576
• 288
• 72

A large study finds a small effect with a tiny p-value. What helps judge practical importance?

• Compare the p-value alone with a smaller significance threshold
• Compare the achieved sample size alone with the planned enrollment
• Compare the estimate and interval with a useful-effect threshold

A design has low power for a small true effect. Only its significant estimates are reported. Which concern follows under common sampling models?

• The selected estimates must have smaller sampling errors than all unreported estimates
• Selection can inflate the reported magnitudes
• The significant subset estimates the original design's power without knowing the true effect

For independent normal observations with known sigma, plan an upper-tailed mean test at alpha = 0.05 and power = 0.80. Use n = ceil[((1.645 + 0.842) sigma / delta)^2], where ceil rounds up. With sigma = 10 and positive target shift delta = 2, what n results?

• 155
• 154
• 156

Use n = ceil[(C / delta)^2], where C > 0 is fixed, delta is a positive target effect and ceil rounds up. A replication substitutes an inflated selected estimate for delta. What sample-size risk follows for the smaller true effect?

• The larger target requires a larger sample at the same alpha and power
• The target changes alpha automatically, leaving required sample size unchanged
• The sample can be too small for the true effect