Courses and Certificates Applied-Probability-and-Statistics Dumps PDF

Variance measures the average squared distance of data values from the mean. For the dataset 2, 4, 6, 8, 10, the mean is (2 + 4 + 6 + 8 + 10) ÷ 5 = 30 ÷ 5 = 6. The deviations from the mean are −4, −2, 0, 2, and 4. Squaring these deviations gives 16, 4, 0, 4, and 16. The sum of squared deviations is 40. Using the sample variance formula, divide by n − 1, where n = 5. Thus, sample variance = 40 ÷ 4 = 10. Option A is correct. Option C, 6.25, does not result from the standard sample variance computation for these values. The important distinction is whether a problem is using sample variance or population variance; the provided answer set and calculation use the sample variance formula. Study Guide references/topics: variance, mean, squared deviations, sample variance.

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This is an observational study because the scientists collect existing information about individuals’ smoking habits and lung cancer diagnoses without assigning treatments or controlling exposure. In an experimental study, researchers impose a condition, treatment, or intervention and then measure the resulting effect. Here, the researchers are not requiring some individuals to smoke and others not to smoke; they are simply recording naturally occurring characteristics and outcomes. The purpose is to examine association, or correlation, between two variables: smoking status and lung cancer diagnosis. Both variables are categorical, so the analysis would likely involve proportions, conditional percentages, or contingency tables. Option A is incorrect because analyzing correlations does not make a study experimental. Option B is incorrect because “testing effects” would require manipulation or assignment. Option C identifies a true feature, a large sample size, but sample size does not determine study type. References/topics from the Study Guide: observational studies, experimental studies, association, categorical variables.

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The sample mean is the standard point estimate for the population mean. A point estimate is a single statistic calculated from sample data and used to estimate an unknown population parameter. If the parameter of interest is the population mean μ, the corresponding sample statistic is x̄. Therefore, the statement is true. This does not mean the sample mean is guaranteed to equal the population mean exactly; sampling variability can cause sample means to differ from the true population mean. However, the sample mean is unbiased under random sampling and becomes more stable as sample size increases. Confidence intervals expand this idea by placing a margin of error around the sample mean. Option B is incorrect because the sample mean is precisely the conventional point estimator for μ. Options C and D impose invalid restrictions. Study Guide references/topics: sample mean, population mean, point estimate, sampling variability.

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The symbol α represents the significance level of a hypothesis test and is the probability of making a Type I error. A Type I error occurs when the null hypothesis is true but the test rejects it. Therefore, α = 0.05 means there is a 5% risk of rejecting a true null hypothesis under the testing procedure. This is often described as the false positive rate. Option B describes a Type II error, which involves failing to reject a false null hypothesis. Option C is incorrect because a 5% significance level corresponds to a 95% confidence level in related confidence-interval contexts, not 5% confidence. Option D is unrelated; α is not a sample-size percentage. The interpretation must connect α directly to the probability of incorrectly rejecting H₀ when H₀ is actually true. Study Guide references/topics: hypothesis testing, significance level, Type I error, null hypothesis.

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