CHAPTER 15

• Statistical inference cannot fix design flaws like using voluntary response samples or uncontrolled experiments
• Data should be a probability sample or from a randomized experiment
• The margin of error in a confidence interval only covers random sampling error
• Undercoverage, nonresponse, or other biases can be more serious than random sampling error
• Opinion polls often have very high nonresponse rates - about 90%

Conditions for Normality

• The sampling distribution should be approximately Normal
• The shape of the population distribution is rarely known, so using sample data to check for outliers is important

Confidence Intervals

• Estimate population parameters with a built-in precision estimate
• The basic form is: Estimate ± Margin of Error
• Requires simple random sampling and the central limit theorem

Confidence Interval Behavior

• Margin of error for the z confidence interval is m = z* σ/√n
• Margin of error decreases if s decreases, confidence decreases, or sample size increases
• Narrower confidence intervals are achieved with larger sample sizes for the same confidence level

Significance Tests

• Statistical significance indicates whether an observed effect is likely to be due to chance because of random sampling
• Statistical significance does not reveal the magnitude of the effect
• Large sample sizes can make even small effects significant
• A drug can lower temperature by 0.4° Celsius (P-value < 0.01), but clinical benefits need a 1°C decrease or greater
• Larger random samples reduce chance variation, so small population effects can be highly significant
• Smaller random samples allow more chance variation, so large population effects can fail to be significant

Multiple Analyses

• The probability of incorrectly rejecting H₀ (Type I error) is the significance level α.
• With α = 5% and multiple analyses, expect a Type I error about 5% of the time
• If you try the same analysis with 100 random samples, about 5 will be significant even if H₀ is true
• For a significance level α = 5%, running multiple tests means an individual result may be significant by chance

Cell Phones and Brain Cancer

• One study found no statistically significant association between cell phone use and brain cancer (glioma)
• When 20 types of glioma were considered separately, an association was found with one rare form
• Greater cell phone use was strangely associated with decreased risk

Mammary Artery Ligation

• Angina, is pain from inadequate blood supply to the heart, may be relieved by ligating mammary arteries
• An uncontrolled experiment reported a statistically significant reduction in angina pain
• The reduction in pain might only be a placebo effect
• A randomized comparative experiment found that ligation was no more effective than a placebo
• Statistical significance indicates something other than chance, but doesn't specify what

Planning Studies

• A particular margin of error might be needed (e.g., drug trial, manufacturing specs)
• The population variability (s) is often fixed, but the number of measurements (n) can be chosen
• It can be determined what sample size is needed to obtain a desired margin of error
• m = z* σ/√n ↔ n = (z* σ / m)²

Calculating Sample Size

• The density of bacteria in solution measured.
• Equipment gives results that vary Normally with standard deviation σ = 1 million bacteria/mL fluid
• Question: How many measurements should be made to obtain a margin of error of at most 0.5 million bacteria/mL with a confidence level of 90%?
• For a 90% confidence interval, z*= 1.645

Power of a Test

• The power of a test of hypothesis is its ability to detect a specified effect size (reject H₀ when a given Hₐ is true) at significance level α
• The specified effect size is chosen to represent a biologically/practically meaningful magnitude

Factors Affecting Power

• The size of the specified effect
• The significance level α
• The sample size n
• The population variance σ²

Power Calculation Example

• Question: do poor mothers have smaller babies?
• National average birth weight is 120 oz: N(µnatl =120, σ = 24 oz)
• Goal: be able to detect an average birth weight of 114 oz (5% lower than the national average)
• Result: an SRS of of 100 babies born of poor mothers will have a significance level of 0.05 with → 80% power

Type I and Type II Errors

• Statistical conclusions are not certain
• A Type I error occurs when we reject the null hypothesis but the null hypothesis is actually true
• A Type II error occurs when we fali to reject the null hypothesis but the null hypothesis is actually false
• The probability of making a Type I error is the significance level α
• The probability of making a Type II error is labeled β, a computed value that depends on a number of factors
• The power of a test is defined as the value 1 – β
• A Type II error is not definitive, because "failing to reject the null hypothesis" does not imply that the null hypothesis is true

Type I Error Example

• A regulatory agency checks air quality for evidence of levels (> 5.0 ppt) of nitrogen oxide (NOx)
• The agency gathers NOx concentrations in an urban area on a random sample of 60 different days and calculates a test of significance to assess whether the mean level of NOx is greater than 5.0 ppt
• A Type I error here would be to believe that the population mean NOx level exceeds 5.0 ppt when it really doesn't