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Statistics: Confidence Intervals and Hypothesis Testing

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  • What is the purpose of a confidence interval in statistics?
    A confidence interval estimates the range within which a population parameter lies with a certain level of confidence, based on sample data.
  • How do you calculate a confidence interval for a population mean with known population standard deviation?
    Use the formula \(\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, z_{\(\alpha\)/2} is the z-score for the confidence level, \(\sigma\) is population standard deviation, and n is sample size.
  • What is the difference between a 90% and a 95% confidence interval?
    A 95% confidence interval is wider than a 90% interval because it requires more certainty, using a larger z-value, thus covering more possible values.
  • What does the confidence level represent?
    The confidence level represents the proportion of times the confidence interval would contain the true population parameter if the experiment were repeated many times.
  • How is the margin of error calculated in a confidence interval?
    Margin of error = \(z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}\), combining the critical value and standard error.
  • What is the formula for the standard error of the mean?
    Standard error = \(\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is population standard deviation and n is sample size.
  • How do you interpret a 95% confidence interval of (2.14, 2.46)?
    We are 95% confident that the true population mean lies between 2.14 and 2.46.
  • What is the relationship between sample size and confidence interval width?
    Increasing sample size decreases the standard error, which narrows the confidence interval, making estimates more precise.
  • What is the null hypothesis in hypothesis testing?
    The null hypothesis (H0) is a statement of no effect or no difference, which we test against the alternative hypothesis.
  • How do you calculate the test statistic for a population mean when population standard deviation is known?
    Test statistic z = \(\frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}\), where \(\mu\)_0 is the hypothesized mean.
  • What does a p-value represent in hypothesis testing?
    The p-value is the probability of obtaining a test statistic as extreme as the observed one, assuming the null hypothesis is true.
  • When do you reject the null hypothesis based on the p-value?
    Reject H0 if the p-value is less than the significance level \(\alpha\) (e.g., 0.05), indicating strong evidence against H0.
  • What is the significance level \(\alpha\) in hypothesis testing?
    The significance level \(\alpha\) is the threshold probability for rejecting the null hypothesis, commonly set at 0.05.
  • How do you interpret a 99% confidence interval compared to a 95% interval?
    A 99% confidence interval is wider than a 95% interval, reflecting greater confidence but less precision.
  • What is the effect of increasing the confidence level on the margin of error?
    Increasing the confidence level increases the critical value z_{\(\alpha\)/2}, which increases the margin of error and widens the interval.
  • What is the formula for the confidence interval for a population proportion?
    Confidence interval = \(\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) is sample proportion.
  • What assumptions are needed to construct a confidence interval for a population mean?
    The sample should be random, the population standard deviation known, and the population distribution normal or sample size large (Central Limit Theorem).
  • How do you interpret a confidence interval that includes the null hypothesis value in hypothesis testing?
    If the confidence interval includes the null hypothesis value, we do not reject H0 at the corresponding confidence level.
  • What is the relationship between confidence intervals and hypothesis tests?
    A two-sided hypothesis test at significance level \(\alpha\) corresponds to a (1-\(\alpha\)) confidence interval; if the null value lies outside the interval, reject H0.
  • What is the critical value z_{\(\alpha\)/2} for a 95% confidence interval?
    The critical value z_{\(\alpha\)/2} for 95% confidence is approximately 1.96.