Statistics: Independent Groups T-Test Concepts
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Questions and Answers

What is the nature of the dependent variable in a between-subjects t-test?

  • Qualitative and ordinal
  • Quantitative and interval-like (correct)
  • Quantitative and categorical
  • Qualitative and nominal

    • In the context of an independent groups t-test, what is the condition regarding the independent variable?

  • It must divide participants into exactly two groups. (correct)
  • It must have more than two levels.
  • It must be qualitative only.
  • It must divide participants into three groups.

    • What would be an appropriate null hypothesis in a t-test comparing two means?

  • H0: μ1 < μ2
  • H0: μ1 = μ2 (correct)
  • H0: μ1 > μ2
  • H0: μ1 ≠ μ2

    • If the noise exposure study mentions H1: μ1 > μ2, what type of hypothesis is this?

    <p>One-sided alternative hypothesis (B)</p> Signup and view all the answers

    What must be true about the populations from which samples are drawn under the null hypothesis?

    <p>They have the same mean and variance. (D)</p> Signup and view all the answers

    In the noise exposure study, what type of independent variable is represented by noise exposure?

    <p>Qualitative and nominal (A)</p> Signup and view all the answers

    Which of the following is an essential characteristic of a true experiment in the context of an independent groups t-test?

    <p>Participants are randomly assigned to groups. (B)</p> Signup and view all the answers

    Why would a researcher use a one-sided alternative hypothesis?

    <p>When the researcher believes one mean will be greater than the other. (B)</p> Signup and view all the answers

    What happens to the power of a test as the effect size increases?

    <p>The power of the test increases. (A)</p> Signup and view all the answers

    What is the relationship between sample size and standard error?

    <p>As sample size increases, standard error decreases. (A)</p> Signup and view all the answers

    In which situation can the power of a test be effectively increased?

    <p>By increasing the sample size. (D)</p> Signup and view all the answers

    What is the implication of a high power in hypothesis testing?

    <p>A high probability of correctly rejecting the null hypothesis. (A)</p> Signup and view all the answers

    Which of the following describes the effect of a larger sample size on statistical significance?

    <p>It increases the likelihood of achieving statistical significance. (C)</p> Signup and view all the answers

    What effect does increasing the effect size have on the probabilities 1-β and β?

    <p>1-β increases and β decreases. (C)</p> Signup and view all the answers

    Which of the following steps is NOT part of the hypothesis testing process?

    <p>Request additional data after testing. (B)</p> Signup and view all the answers

    How does the shape of the distribution change with an increase in sample size?

    <p>The distribution becomes taller. (C)</p> Signup and view all the answers

    What is the null hypothesis when comparing two population means?

    <p>μ1 - μ2 = 0 (C)</p> Signup and view all the answers

    What do you need to do when you do not know the population standard deviation (σ) in a two-sample case?

    <p>Estimate it using the two samples’ standard deviations (B)</p> Signup and view all the answers

    What is the purpose of pooling the variance estimates in the context of a t-test?

    <p>To provide a pooled estimate of the parameter value (B)</p> Signup and view all the answers

    How is the estimated standard error of the difference calculated?

    <p>By replacing σwith s^pooled in the formula (B)</p> Signup and view all the answers

    What does the formula for the independent samples t-test require?

    <p>Estimation of sample variances (A)</p> Signup and view all the answers

    Why is the pooled standard deviation (s^pooled) important in hypothesis testing?

    <p>It provides a more accurate measure of variability between samples (C)</p> Signup and view all the answers

    What does the degrees of freedom (df) represent in the pooled variance formula?

    <p>The sum of each sample’s df (D)</p> Signup and view all the answers

    What happens if your null hypothesis (H0) assumes a different value than μ1 = μ2?

    <p>You should replace the value with your assumption (D)</p> Signup and view all the answers

    What effect does a small sample size have on confidence intervals?

    <p>It increases the confidence interval width. (B)</p> Signup and view all the answers

    What is the t-score used to calculate a 99% confidence interval for a sample of 5 cars?

    <p>4.604 (C)</p> Signup and view all the answers

    If the mean safety rating for a sample of 20 cars is still 8, what would the confidence interval be?

    <p>7.40 to 8.60 (B)</p> Signup and view all the answers

    What must be done when the population standard deviation is not known?

    <p>Estimate the population standard deviation. (A)</p> Signup and view all the answers

    What is the confidence interval calculated using a mean of 8 and a standard deviation of 0.94 for a sample of 5 cars?

    <p>6.06 to 9.94 (C)</p> Signup and view all the answers

    When increasing the number of tested vehicles, what is the impact on the confidence interval?

    <p>It becomes narrower. (B)</p> Signup and view all the answers

    What is the null hypothesis stated in this context?

    <p>The mean score of the class is equal to the population mean. (C)</p> Signup and view all the answers

    What does a confidence interval with a level of 99% indicate?

    <p>There is a 1% chance the true mean is outside the interval. (D)</p> Signup and view all the answers

    What is the value of the critical z-scores for a two-sided test at an alpha level of 0.05?

    <p>-1.96 and +1.96 (A)</p> Signup and view all the answers

    What does a p-value of 0.0096 indicate in this scenario?

    <p>There is strong evidence to reject the null hypothesis. (C)</p> Signup and view all the answers

    How is the formula for the confidence interval constructed?

    <p>Y ± (t x s / √N) (B)</p> Signup and view all the answers

    What does the area under the standard normal curve from negative infinity to -2.59 represent?

    <p>The cumulative distribution function value for z = -2.59. (C)</p> Signup and view all the answers

    What conclusion is drawn when the z-score is -2.59 in relation to the null hypothesis?

    <p>The class is significantly different from the general population. (C)</p> Signup and view all the answers

    What is the probability of observing a score of 94 or less based on random selection?

    <p>0.004 or 0.4% (D)</p> Signup and view all the answers

    How is the mean of the sampling distribution calculated?

    <p>By adding up all sample means and dividing by the number of means. (B)</p> Signup and view all the answers

    What interpretation can be made if the area under the standard normal curve from negative infinity to -0.4 is 0.345?

    <p>34.5% of scores fall below -0.4. (A)</p> Signup and view all the answers

    What is the effect of increasing the sample size on the confidence interval?

    <p>The confidence interval becomes smaller. (C)</p> Signup and view all the answers

    For a sample size of n = 12, what is the critical t-value for a 95% confidence interval?

    <p>2.201 (D)</p> Signup and view all the answers

    What does a population mean that falls outside the confidence interval suggest?

    <p>It indicates a statistically significant result. (A)</p> Signup and view all the answers

    Which statement is true regarding the critical t-score as the sample size increases?

    <p>The critical t-score decreases. (A)</p> Signup and view all the answers

    What is the critical t-value for a sample size of n = 26 at a 95% confidence level?

    <p>2.060 (D)</p> Signup and view all the answers

    What happens to the confidence interval around the sample mean as the sample size decreases?

    <p>It becomes larger. (B)</p> Signup and view all the answers

    For a 99% confidence interval, what is the lower limit if the sample mean is 83.00 and the critical t-value is 2.756 with a standard error of 3.17?

    <p>74.26 (B)</p> Signup and view all the answers

    Why is it important to consider degrees of freedom when using the t-distribution?

    <p>It affects the shape of the t-distribution. (C)</p> Signup and view all the answers

    At an alpha level of 0.01, if the population mean is outside the confidence interval, what can be concluded?

    <p>Reject the null hypothesis. (D)</p> Signup and view all the answers

    What does the symbol $ŝ$ represent in the confidence interval formula?

    <p>Standard deviation of the sample. (D)</p> Signup and view all the answers

    In a two-tailed test, what is the total probability level associated with a 95% confidence interval?

    <p>0.05 (B)</p> Signup and view all the answers

    How does a 99% confidence interval compare to a 95% confidence interval in context?

    <p>99% is wider than 95%. (D)</p> Signup and view all the answers

    What is indicated by a critical t-value of 1.645?

    <p>It pertains to a 90% confidence level. (B)</p> Signup and view all the answers

    Study Notes

    Hypothesis Testing: Correct Decisions and Errors

    • In any hypothesis test, there are four possible outcomes
    • The true situation: Either the null hypothesis is actually true or the alternative hypothesis is actually true
    • The decision is either not to reject the null hypothesis or reject the null hypothesis and accept the alternative hypothesis

    Hypothesis Testing: Correct Decisions and Errors

    • The True Situation column represents whether the null hypothesis (H0) or the alternative hypothesis (HA) is true
    • The Decision column represents the decision made in the hypothesis test (do not reject H0 or reject H0 and accept HA)
    • Correct Decision: The decision matches the true situation
    • Type I Error: Rejecting H0 when it is actually true
    • Type II Error: Failing to reject H0 when it is actually false

    Hypothesis Testing: Correct Decisions and Errors

    • Probabilities for each outcome are represented as conditional probabilities
    • The probabilities are conditional on the true situation
    • P(Not Rejecting H0 when H0 is True) = 1 − α
    • P(Rejecting H0 when H0 is True) = α
    • P(Not Rejecting H0 when HA is True) = β
    • P(Rejecting H0 when HA is True) = 1 − β

    Significance Level

    • The probability of a Type I error is represented by the symbol α and is called the significance level
    • The probability of a correct rejection of the null hypothesis in favor of the alternative hypothesis is represented by 1 − β and is called the power of the test

    Type II Error and the Power of the Test

    • A standardized ability test has scores that are normally distributed with μ = 500 and σ = 100
    • An investigator is interested in the effects of CAI (Computer Assisted Instruction) on the performance of 25 students
    • The investigator predicts that CAI will lead to an improvement in performance, i.e., HA: μ = 550

    Type II Error and the Power of the Test

    • From the sampling distribution under the null hypothesis (with a 5% significance level)
    • Critical value is 1.645
    • If the observed mean is 520.4, the z-score would be 1.02 (calculated by (520.4 - 500) / 20)
    • Fail to reject the null hypothesis because z-score is outside rejection region

    Power of the Test: Effect Size

    • As the effect size increases in an example (from μ = 550 to μ = 580), the power of the test increases, i.e., the probability of a correct rejection of H0 increases and the probability of a Type II error decreases
    • By increasing sample sizes, standard error from estimating population parameter (mean) decreases, and thus the power of the test increases

    The Power of the Test: Effect Size

    • With reference to the CAI example, imagine that the investigator had tested 100 students (n = 100 instead of 25)
    • As the sample size increases, the standard error decreases, and the distribution becomes narrower, which increases the power to detect the effect
    • Small differences in means can still be statistically significant if the sample size is large enough

    Hypothesis Testing Recap

    • Begin with a null and alternative hypothesis
    • Set the alpha level
    • Perform the appropriate statistical test
    • Calculate the p-value
    • Compare the p-value to the alpha level
    • Alternatively, find the critical and observed test statistics for a decision

    Sampling Distributions

    • A population of scores reflects the frequency of occurrence of every score in the distribution
    • Frequency distributions are also called probability distributions
    • Given the mean and standard deviation of a normal distribution, probability statements can be made about the likelihood of selecting a score from a specified area of the distribution

    Sampling Distributions

    • In hypothesis testing, you're usually interested in the means rather than individual scores
    • To make probability statements about a randomly selected mean falling within a specified area under the normal curve, you need a normal distribution of means

    Sampling Distributions

    • Mean of the sampling distribution is exactly the same as the mean of the population (μ = μy)
    • Standard deviation of the sampling distribution (σy) = σ/√n
    • Variability of the sampling distribution is determined by: The variability of the population distribution, and the size of the samples used to establish the sampling distribution
    • As sample size increases, the sampling distribution approaches a normal distribution (central limit theorem)
    • A large sample (N > 30) is usually sufficient for the sampling distribution to be normally distributed, regardless of the population distribution

    One-Sample t-Test

    • When using a small sample, the sample standard deviation (s) is used as an estimate of the population standard deviation (σ)

    Student's t Distribution

    • When working with small samples, there's more error in estimating parameters based on this information
    • The formula for the t-statistic (t) is t = (Υ - μy)/Sy and calculates estimated number of standard errors that the sample mean is from μ
    • In a real data study, you generally have little control over the effect size, and more control over the sample sizes
    • A similar value of t-statistic is observed when comparing with z-score if the sample size is larger than 40

    Student's t Distribution

    • The symbol v is used to represent degrees of freedom. A particular t-value always has a df associated with it: t(df)
    • For a one-sample case, df = N-1
    • You cannot use a z-table; must use a different table, the 't-distribution table'

    z Distribution vs. t Distribution

    • In a z-distribution, population standard deviation (σ) is known while in a t-distribution it is not known

    Example

    • Adults are recommended to exercise at least twice a week
    • Sample of 35-40-year-olds consists of 30 individuals, with mean 1.84 and estimated standard deviation 1.68
    • The viablity of hypothesis that μ=2 is examined using a one sample t-test

    Example

    • The critical values of t for a non-directional test (alpha = 0.05) and 29 degrees of freedom are ±2.045
    • The test statistic for the one sample t-test is t(29) = -0.52
    • Reject/Fail to reject the null hypothesis: Because the value of the t-statistic (-0.52) does not exceed the negative critical value (-2.045), we fail to reject the null hypothesis

    Example (Confidence Intervals Approach)

    • The critical values of t for a non-directional test (alpha = 0.05) and 29 degrees of freedom are ±2.045
    • Calculated standard error = 0.31
    • One-sample CI at 95% is between 1.21 and 2.47
    • The null hypothesis that μ=2 is not rejected as the population mean lies in the interval

    Example

    • A researcher conducted a reading test on 30 students with a mean of 83 and standard deviation estimate of 17.35
    • 95% confidence interval is 76.52 to 89.48
    • 99% confidence interval is 74.26 to 91.94

    Example: Sample Size and t-Distribution

    • For a sample of size 12 (df=11), 95%CI critical t-value (or t-score) is 2.201 for a two-sided test
    • With increased sample size (e.g., 26, df=25), the critical t-score becomes 2.060, illustrating an inverse relationship between sample size, standard error, and thus the confidence interval

    Example: Independent samples t tests

    • A researcher examines whether a new teaching method affects student math test scores compared to a traditional method
    • Group 1 (traditional method): n1 = 10, X₁ = 75, ŝ1 = 8
    • Group 2 (new method): n2 = 12, X2 = 82, ŝ2 = 10
    • Significance level = 0.05 (two-sided test)
    • Null Hypothesis: The teaching method has no effect on math scores (H0: μ1 = μ2)
    • Alternative Hypothesis: The teaching method affects math scores (H1: μ1 ≠ μ2)
    • Pooled Standard Deviation: 9.15
    • Estimated Standard Error of Mean Difference: 3.92
    • Calculated t-Statistic: -1.79
    • Critical t-value: ± 2.086
    • Fail to reject null hypothesis: The calculated t-value is not outside the critical t-value range

    Assumptions of the Independent Groups t-test

    • Samples are independently selected from their respective populations
    • Scores each population are normally distributed
    • Dependent variable is at least at interval level of measure
    • Scores in two populations have equal variances

    Using Independent Samples t-Test

    • You are interested in the effects of physical exercise on health. You expect that people who regularly exercise will spend less days sick within a year than non-exercising people.
    • Exercising group: n1 = 5, X1 = 24, s1 = 4.5
    • Non-exercising group: n2 = 15, X2 = 30, s2 = 9.4
    • Significance level: .05
    • Null Hypothesis (H0): μ1 = μ2
    • Alternative Hypothesis (H1): μ1 < μ2
    • Degrees of Freedom: df=18
    • Critical t-value: -1.734
    • Calculated t-statistic: -1.36

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    This quiz tests your understanding of the independent groups t-test and dependent variables. The questions explore hypotheses, power of tests, and the characteristics of true experiments. It's essential for anyone studying statistics or conducting experiments.

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