The Null Hypothesis
In hypothesis testing, what does the symbol H₀ represent?
A: The null hypothesis
What does a critical value define in hypothesis testing?
A: The boundary between the rejection and non-rejection regions
What is the difference between a one-tailed and a two-tailed test?
A: A one-tailed test looks for an effect in only one direction, while a two-tailed test looks for an effect in either direction
What does a 95% confidence interval tell us?
A: We are 95% confident that the true population parameter falls within this interval
When should you use a t-test instead of a z-test?
A: When the population standard deviation is unknown and/or the sample size is small
The null hypothesis typically contains what type of comparison operator?
A: An equals sign (=) or "no difference" or "no effect"
For a 95% confidence level in a two-tailed z-test, what is the critical z-value?
A: ±1.96
If your alternative hypothesis is that the mean is greater than a specific value, what type of test would you use?
A: A right-tailed (or upper-tailed) test
How does the width of a confidence interval change as the sample size increases?
A: The confidence interval becomes narrower as sample size increases
What does the "degrees of freedom" refer to in a t-test?
A: For a one-sample t-test, it's the sample size minus one (n-1)
What is the statistical term for incorrectly rejecting a true null hypothesis?
A: Type I error
How does the critical value change if you increase your significance level from 0.05 to 0.10?
A: The critical value decreases (becomes less extreme)
If α = 0.05 for a two-tailed test, what is the α value for each tail?
A: 0.025 in each tail
If you want to create a confidence interval with the same width but a higher confidence level (e.g., from 95% to 99%), what must you do to your sample size?
A: Increase the sample size
What assumption must be met regarding the data distribution when performing a t-test?
A: The data should be approximately normally distributed
If your null hypothesis states "there is no significant difference between the mean test scores of male and female students," what would be the alternative hypothesis?
A: There is a significant difference between the mean test scores of male and female students
In a right-tailed hypothesis test with a significance level of 0.01, what is the critical z-value?
A: 2.33
When testing if a new teaching method is either better or worse than the traditional method, would you use a one-tailed or two-tailed test? Explain why.
A: A two-tailed test, because we're interested in detecting a difference in either direction (better or worse)
The margin of error for a 95% confidence interval for a population mean is 3.5. What is the width of this confidence interval?
A: 7 units (twice the margin of error)
Explain the difference between a paired t-test and an independent samples t-test.
A: A paired t-test is used when the samples are dependent (like before/after measurements on the same subjects), while an independent samples t-test is used when the two samples are unrelated to each other
In a clinical trial testing a new drug against a placebo, the researchers want to show the drug is effective. Explain why the null hypothesis would be set as "the drug has no effect" rather than "the drug is effective."
A: The null hypothesis is set as "the drug has no effect" because in hypothesis testing, we aim to reject the null hypothesis. This approach requires evidence to overturn the presumption of no effect, placing the burden of proof on showing that the drug works, which is more scientifically rigorous than assuming effectiveness.
researcher conducts a hypothesis test with a t-distribution with 24 degrees of freedom and a significance level of 0.05 for a two-tailed test. Calculate the critical t-value and explain how this value would change if the sample size increased to 100.
A: The critical t-value is approximately ±2.064. If the sample size increased to 100, the degrees of freedom would increase to 99, and the critical t-value would decrease to approximately ±1.984, becoming closer to the z-value of ±1.96 as the t-distribution approaches the normal distribution with larger sample sizes.
A quality control engineer is testing the breaking strength of cables. The manufacturer claims the cables have a mean breaking strength of 500 pounds. Under what specific circumstances should the engineer use a one-tailed test instead of a two-tailed test, and which direction should the tail be? Provide statistical reasoning for your answer.
A: The engineer should use a one-tailed test if they're only concerned about the cables being weaker than claimed (a left-tailed test), which would be appropriate if only weaker cables pose a safety risk. Alternatively, if they're testing whether a new manufacturing process made the cables stronger, they would use a right-tailed test. The choice depends on whether the practical concern is unidirectional and established before data collection, as using a one-tailed test provides more power to detect an effect in the specified direction.
A random sample of 40 students had a mean exam score of 75 with a standard deviation of 8. Calculate the 95% confidence interval for the population mean exam score. Then explain how the interpretation of this interval would differ from a 95% prediction interval for an individual student's score.
A: The 95% confidence interval is approximately 75 ± 2.48, or [72.52, 77.48]. A confidence interval estimates where the population mean lies, while a prediction interval would be much wider because it aims to capture where an individual student's score would fall, accounting for both the uncertainty in estimating the mean and the natural variability of individual scores around that mean.
A researcher wants to compare the effectiveness of three different study methods. The t-test would not be appropriate for this analysis. Explain why, and describe what statistical test would be more appropriate, including the key assumptions that would need to be verified for that test.
A: The t-test is not appropriate because it can only compare two groups, not three. The researcher should use a one-way ANOVA (Analysis of Variance), which can compare means across multiple groups. Key assumptions for ANOVA include: (1) independent observations, (2) normal distribution of the data in each group, and (3) homogeneity of variances across groups. If these assumptions are violated, non-parametric alternatives like the Kruskal-Wallis test might be needed.