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Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Sampling Distributions
• A sampling distribution of the mean is the distribution of an infinite number of sample means from the population, for samples of a given size (e.g., 25)
• A sampling distribution is a theoretical (not an actual) distribution of values
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
SEMs and Probabilities
• With a sample IQ mean of 101.0 (the estimated population mean), and an SEM = 2.1, we could construct a distribution around the mean to estimate the likelihood of obtaining a sample mean within a given range– Assumption: Scores are normally
distributed, and thus we know how many SDs are above/below the mean
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
SEMs and Probabilities (cont’d)
• In this scenario, the probability is about .95 that all sample IQ means from the population of 10-year-old children would be no more than two SDs below and two SDs above the mean of 101.0 (i.e, between 96.8 and 105.2)– With a sample of 225 (SEM = 0.7), the range
would be between 99.6 and 102.4– Larger sample size Greater precision in
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Confidence Intervals
• A 95% confidence interval (95% CI) designates the range of values within which the parameter has a 95% probability of lying
• Constructing a CI involves calculating confidence limits (the upper and lower limit of what is probable, at the specified probability level)– The most commonly reported CIs are 95%
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Confidence Limits
• The formula for confidence limits around a mean involves three components: – The sample mean– The estimated SEM– The value corresponding to the area from a
theoretical distribution for the desired CI (e.g., 95%)
• For sample means, the appropriate theoretical distribution is the t distribution
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Calculation of CIs Around Means
• Formula for a 95% CI around a mean:95% CI = M ± (t × SEM)
• Earlier example of IQs: N = 225 children, M = 101.0, SEM = 0.7, and t (from a theoretical table) = 1.96, which is more accurate than earlier estimate of 2.0
• 95% CI = 101.0 (99.63, 102.37) – There is a 95% probability that the true
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Hypothesis Testing
• Hypothesis testing (second broad approach to statistical inference) uses laws of probability to help researchers make objective decisions about accepting or rejecting a null hypothesis
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Research Hypotheses
• The null hypothesis contrasts with researchers’ actual research hypothesis, which typically states a prediction that variables in the study ARE related, e.g.:– Cigarette smoking is related to lung cancer– Turning patients is related to the incidence of
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Hypothesis Testing Logic
• Hypothesis testing is similar to English-based criminal justice system– The accused is assumed to be innocent
• Variables are assumed to be “innocent” of any relationship (i.e., the null hypothesis is assumed to be true) until evidence from sample data indicate a high probability that the null is NOT true
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Type I and Type II Errors
• Two types of error in statistical decisions:– Type I error: The null hypothesis is really true
in the population, but the researcher rejects it (a false positive) E.g., an ineffective intervention is erroneously
considered effective
– Type II error: The null hypothesis is really false in the population, but the researcher accepts it (a false negative) E.g., an effective intervention is erroneously
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Controlling Type I Error Risk
• Type I errors are controlled through the level of significance, the probability accepted as the risk of a false positive– The level of significance or alpha (α) is the
area in the theoretical probability distribution corresponding to a rejection of the null hypothesis
– In most cases, α = .05, which is a 5% risk of a Type I error; this is analogous to a 95% CI
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Critical Regions
• Researchers calculate a test statistic using their sample data
• They reject the null hypothesis if the test statistic falls at or beyond a critical region on the theoretical distribution for their test statistic; they accept the null hypothesis otherwise
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
One-Sample t Tests
• To illustrate statistical tests and critical regions, consider the one-sample t-test, a statistical test that tests the null hypothesis that the population mean is a specific value
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Statistical Significance
• When the null hypothesis is rejected, the results are statistically significant
• If the null hypothesis is retained (whenever p > .05), the results are statistically nonsignificant– A nonsignificant result reflects an outcome that
could have been obtained as a result of chance more than five out of 100 times
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Meaning of Statistical Significance
• A statistically significant result is one that has a high probability of being “real” in the population, and probably does not merely reflect a chance fluctuation
• Statistical significance does not mean the result is important, relevant, or clinically meaningful
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Two-Tailed Statistical Tests
• A two-tailed test is one that uses both tails of a sampling distribution to determine the critical region (the region for rejecting the null hypothesis)
• In our example of adolescents’ neutrality to a low-carb diet, we had a nondirectional alternative hypothesis (μ ≠ 5.0) that implied the need to look at both tails
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
One-Tailed Statistical Tests
• A one-tailed test is one that uses only one tail of a sampling distribution in determining the critical region
• A one-tailed test may be appropriate if the alternative hypothesis is directional
• For example, we might hypothesize that adolescents, on average, would have positive attitudes toward a low-carb diet (μ > 5.0), suggesting we look only in the upper tail
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Use of One-Tailed Tests
• Two-tailed tests are the norm
• Two-tailed tests are more conservative (have less statistical power) than one-tailed tests, but researchers should have a strong justification for looking in only one tail
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Parametric Versus Nonparametric (cont’d)
• Some purists insist on nonparametric tests when assumptions are violated, but because parametric tests are robust and more powerful they are often preferred when violations of assumptions are modest