The null hypothesis (H₀) states that there is no difference between the two population means.
True
Steps to identify the research question and formulate hypotheses
1️⃣ Define the two populations and their means
2️⃣ Determine whether you are testing for a difference, increase, or decrease
3️⃣ State the null and alternative hypotheses accordingly
The two samples in a two-sample t-test must be randomly selected from their respective populations.
The two samples in a two-sample t-test must be independent of each other.
True
The assumption of randomness in an independent samples t-test ensures that the samples are representative and minimizes bias.
True
What are the key assumptions for the independent samples t-test?
Randomness, normality, independence
Why is independence required between the samples in an independent samples t-test?
Avoids correlated errors
The research question in a two-sample t-test is typically framed as: "Is there a difference between the means of two populations
Rejecting the null hypothesis means accepting the alternative hypothesis.
True
What theorem can compensate for non-normality if sample sizes are large enough?
Central Limit Theorem
The normality condition is necessary for accurate p-value calculation.
If sample sizes are large enough (n ≥ 30), the Central Limit Theorem can compensate for non-normality.
The randomness assumption for the independent samples t-test ensures the samples are representative.
The significance level, denoted as alpha (α), is the probability threshold for rejecting the null hypothesis.
The degrees of freedom for an independent samples t-test can be calculated using statistical software.
Steps to calculate the test statistic for an independent samples t-test
1️⃣ Calculate the sample means for each group
2️⃣ Calculate the sample variances for each group
3️⃣ Plug the values into the test statistic formula
One of the steps in calculating the t-statistic is to calculate the sample variances for each group.
True
The research question in a test for the difference of two population means is typically in the form: "Is there a difference between the means of two populations?"
Why is it crucial to formulate hypotheses correctly when testing for the difference of two population means?
To reject the null hypothesis
What is the alternative hypothesis in the example "Is there a significant difference in test scores between students who use online learning resources and those who do not?"
μ₁ ≠ μ₂
What theorem can compensate for non-normality if sample sizes are large enough in a two-sample t-test?
Central Limit Theorem
The independent samples t-test is used to determine if there is a significant difference between the means of two groups.
The independent samples t-test is used to determine if there is a significant difference between the means
The normality assumption for the independent samples t-test is necessary for accurate p-value
Match the assumptions with their descriptions:
Randomness ↔️ Samples are randomly selected
Normality ↔️ Sample distributions are normal
Independence ↔️ Samples are independent of each other
What does the alternative hypothesis state in a two-sample t-test?
There is a difference between means
The normality assumption can be checked using a histogram, normal probability plot, or formal normality tests like the Shapiro-Wilk
What condition ensures representativeness and minimizes bias in sampling?
Randomness
How can the normality assumption be checked for a sample distribution?
Histogram or normal probability plot
Why is independence important in a two-sample t-test?
Avoids correlated errors
In what scenario would an independent samples t-test be appropriate to use?
Comparing test scores between groups
What is a commonly used value for alpha in statistical testing?
0.05
What is the formula for the test statistic in an independent samples t-test?
t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}</latex>
In the formula for the t-statistic, x̄₁ and x̄₂ represent the sample means
In the t-statistic formula, n₁ and n₂ denote the sample sizes
In the example, the sample mean of Group 1 is 85
What is the calculated value of the t-statistic in the example?
4.12
For a two-tailed test, the p-value obtained from a t-distribution table must be multiplied by 2
The hypotheses in a hypothesis test are the proposed answers to the research question.
The alternative hypothesis (H₁) states that there is a difference between the two population means.