Podcast
Questions and Answers
Which of these is the MOST accurate description of the Central Limit Theorem (CLT)?
Which of these is the MOST accurate description of the Central Limit Theorem (CLT)?
- The CLT suggests that larger sample sizes always reduce the standard deviation of the population.
- The CLT implies that the mean of any sample will always equal the true population mean, given a sufficiently large sample.
- The CLT only applies when the population distribution is normal; it doesn't hold for skewed or non-normal distributions.
- The CLT states that the distribution of sample means approaches a normal distribution as sample size increases, regardless of the population's distribution. (correct)
In hypothesis testing, what is the primary role of the null hypothesis ($H_0$)?
In hypothesis testing, what is the primary role of the null hypothesis ($H_0$)?
- It serves as a baseline assumption of no effect or no difference, which the researcher attempts to disprove. (correct)
- It is the hypothesis that the researcher believes to be true and aims to provide evidence for.
- It represents the researcher's prediction about the effect or relationship being studied.
- It defines the level of statistical significance ($\alpha$) required to reject the alternative hypothesis ($H_1$).
What does Cohen's d measure in the context of statistical analysis?
What does Cohen's d measure in the context of statistical analysis?
- The degrees of freedom in a t-test.
- The magnitude of the effect, standardized by the standard deviation. (correct)
- The sample standard deviation.
- The probability of making a Type I error.
Which of the following statements accurately differentiates between Type I and Type II errors?
Which of the following statements accurately differentiates between Type I and Type II errors?
Which factor does NOT typically influence the statistical power of a hypothesis test?
Which factor does NOT typically influence the statistical power of a hypothesis test?
Under what conditions is Student's t-distribution MOST appropriately used instead of the standard normal (z) distribution?
Under what conditions is Student's t-distribution MOST appropriately used instead of the standard normal (z) distribution?
What is the 'degrees of freedom' (df) associated with a one-sample t-test, and how does it influence the shape of the t-distribution?
What is the 'degrees of freedom' (df) associated with a one-sample t-test, and how does it influence the shape of the t-distribution?
What primary assumption MUST be met to accurately conduct a one-sample t-test?
What primary assumption MUST be met to accurately conduct a one-sample t-test?
In the context of hypothesis testing, what is a 'critical value' and how is it used?
In the context of hypothesis testing, what is a 'critical value' and how is it used?
How does increasing the confidence level (e.g., from 95% to 99%) influence the width of a confidence interval?
How does increasing the confidence level (e.g., from 95% to 99%) influence the width of a confidence interval?
Why is it generally inappropriate to 'accept' the null hypothesis ($H_0$) in statistical hypothesis testing?
Why is it generally inappropriate to 'accept' the null hypothesis ($H_0$) in statistical hypothesis testing?
What is the purpose of calculating a 'test statistic' (e.g., z-score, t-statistic) in hypothesis testing?
What is the purpose of calculating a 'test statistic' (e.g., z-score, t-statistic) in hypothesis testing?
The heights of a population of women are normally distributed with a known population mean ($\mu$) of 65 inches and a standard deviation ($\sigma$) of 3.5 inches. A researcher collects a sample of 35 female college basketball players and finds that the mean height of the sample is 67 inches. Determine the standard error of the mean ($\sigma_{\bar{x}}$).
The heights of a population of women are normally distributed with a known population mean ($\mu$) of 65 inches and a standard deviation ($\sigma$) of 3.5 inches. A researcher collects a sample of 35 female college basketball players and finds that the mean height of the sample is 67 inches. Determine the standard error of the mean ($\sigma_{\bar{x}}$).
Using the information of the prior question, what is the calculated z-score ($z_{obs}$) for this sample mean? Recall: $\mu = 65$, $\sigma = 3.5$, $n = 35$, $\bar{x} = 67$ and $\sigma_{\bar{x}} \approx 0.59$.
Using the information of the prior question, what is the calculated z-score ($z_{obs}$) for this sample mean? Recall: $\mu = 65$, $\sigma = 3.5$, $n = 35$, $\bar{x} = 67$ and $\sigma_{\bar{x}} \approx 0.59$.
Given a one-tailed hypothesis test with $\alpha = 0.05$, what is the critical z-value ($z_{crit}$)?
Given a one-tailed hypothesis test with $\alpha = 0.05$, what is the critical z-value ($z_{crit}$)?
In the context of the female basketball player height example, the calculated $z_{obs}$ is 3.39 and the critical z-value ($z_{crit}$) for our one-tailed test when $\alpha = 0.05$ is 1.645. What statistical decision should be made?
In the context of the female basketball player height example, the calculated $z_{obs}$ is 3.39 and the critical z-value ($z_{crit}$) for our one-tailed test when $\alpha = 0.05$ is 1.645. What statistical decision should be made?
What is the correct interpretation of the decision to reject the null hypothesis in the context of the basketball player height example?
What is the correct interpretation of the decision to reject the null hypothesis in the context of the basketball player height example?
For a two-tailed hypothesis test with $\alpha = 0.05$, what are the critical z-values?
For a two-tailed hypothesis test with $\alpha = 0.05$, what are the critical z-values?
If, for a two-tailed test, your calculated test statistic ($z_{obs}$) is -2.0, and the critical z-values ($z_{crit}$) are $\pm$1.96, what decision should be made regarding the null hypothesis?
If, for a two-tailed test, your calculated test statistic ($z_{obs}$) is -2.0, and the critical z-values ($z_{crit}$) are $\pm$1.96, what decision should be made regarding the null hypothesis?
What is the distinction between 'descriptive statistics' and 'inferential statistics'?
What is the distinction between 'descriptive statistics' and 'inferential statistics'?
Which statistical test is MOST appropriate when comparing the means of two independent groups when the population standard deviation is unknown, and the sample sizes are small (e.g., n < 30)?
Which statistical test is MOST appropriate when comparing the means of two independent groups when the population standard deviation is unknown, and the sample sizes are small (e.g., n < 30)?
What is the primary goal of establishing an alpha level ($\alpha$) in hypothesis testing?
What is the primary goal of establishing an alpha level ($\alpha$) in hypothesis testing?
If a researcher decreases the alpha level ($\alpha$), what is the MOST likely consequence regarding Type I and Type II errors?
If a researcher decreases the alpha level ($\alpha$), what is the MOST likely consequence regarding Type I and Type II errors?
Why does the Central Limit Theorem (CLT) play a crucial role in hypothesis testing, particularly when dealing with sample means?
Why does the Central Limit Theorem (CLT) play a crucial role in hypothesis testing, particularly when dealing with sample means?
A researcher aims to determine if a new teaching method improves student test scores, and sets up a one-tailed hypothesis test. What consideration must be made when choosing between a right-tailed and a left-tailed test?
A researcher aims to determine if a new teaching method improves student test scores, and sets up a one-tailed hypothesis test. What consideration must be made when choosing between a right-tailed and a left-tailed test?
What is the benefit of using a confidence interval compared to only performing a hypothesis test?
What is the benefit of using a confidence interval compared to only performing a hypothesis test?
In what step of hypothesis testing do you calculate the standard error of the mean and mark rejection regions?
In what step of hypothesis testing do you calculate the standard error of the mean and mark rejection regions?
What is the next step of hypothesis testing after, "Choose a test statistic?"
What is the next step of hypothesis testing after, "Choose a test statistic?"
A researcher fails to reject the null hypothesis. What is a possible real-world explanation for this statistical conclusion?
A researcher fails to reject the null hypothesis. What is a possible real-world explanation for this statistical conclusion?
Flashcards
Sampling distribution construction
Sampling distribution construction
Plot all possible random sample means of a given size from a population.
Hypothesis testing definition
Hypothesis testing definition
To figure out how likely or unlikely it would be to get a sample mean of a particular size, given a hypothesized population mean (μ).
Steps In Hypothesis Testing
Steps In Hypothesis Testing
State H0 and H1, collect data, establish alpha, choose a test statistic, compute test statistic, make a decision about H0, describe decision in words.
Null hypothesis (H0)
Null hypothesis (H0)
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Alternative hypothesis (H1)
Alternative hypothesis (H1)
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Establish alpha (α)
Establish alpha (α)
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Critical value
Critical value
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Test statistic
Test statistic
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Accepting the null hypothesis
Accepting the null hypothesis
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Study Notes
- Study notes on central limit theorem and hypothesis testing.
Topics Outline
- Hypothesis testing: The one-sample z test.
- Understanding Cohen's d as a measure of effect size.
- Grasping Type I vs. Type II errors.
- Understanding power and what affects it.
- The one-sample t test including:
- t distribution.
- degrees of freedom (df).
- one-sample t test by hand and via SPSS.
- effect size.
- assumptions of the test.
- Confidence intervals
Lecture Outline
- Refresher on Sample Statistics (SS), Population Parameters (PP), and the Central Limit Theorem (CLT)
- Hypothesis Testing involving 8 steps
Sample Statistics vs. Population Parameters
- Plot all possible random sample means of a given size from a population, to construct a sampling distribution of the mean
Hypothesis Testing
- It is used to determine how likely or unlikely it would be to get a sample mean of a particular size, given a hypothesized population mean (μ).
- Focus is on how two or more sample means differ from each other (e.g., treatment vs. control group).
- It is rare to know µ.
Steps in Hypothesis Testing
- State Ho and H₁
- Collect data
- Establish α (usually α = .05)
- Choose a test statistic (e.g., one-sample z test)
- Characterize the sampling distribution, compute mean & SEM of the samp. dist., define Zcrit, and mark rejection regions (Ho, H₁)
- Compute test statistic (e.g., Zobs, tobs, Fobs)
- Make a decision about Ho (reject or fail to reject Ho)
- Describe the decision in words.
Step 1: State the Hypotheses
- Set up the alternative hypothesis (H₁) to "knock down" the null hypothesis.
- The Null hypothesis (H。) is that the sample comes from a population with a mean height of 65".
- Alternative Hypotheses (H₁)
- Two-tailed, non-directional hypothesis: The sample does not come from a population with a mean height of 65"
- One-tailed, directional hypothesis: The sample comes from a population with a mean height greater than 65" (less commonly used)
Step 2: Collect the Data
- Data needs to be collected
Step 3: Establish Alpha (α)
- It is necessary to decide on a cut off for what is "unusual"
- Need to define an unlikely event.
- It is an outcome that would happen less than 5% of the time by chance.
- "Significance level of .05" means:
- Willing to take a 5% risk of making an error in the decision about statistical significance.
- Alpha (the significance level) is a type of error.
- A sample mean height of 67 would be unusual if it occurred less than 5% of the time, given that the null hypothesis is true.
- The Z table is where we can determine if a mean of 67" occurred less than 5% of the time
Refresher: Z Score!
- After looking up p = .0500 on the Z table, the corresponding Z score is called Zcrit
- To conduct the hypothesis test:
- Look up the critical value (Zcrit) using the Z table.
- Calculate a test statistic (Zobs).
- Compare Zobs to Zcrit to determine whether to reject or fail to reject the null hypothesis.
Step 4: Choose a Test Statistic
- This step depends on what information is available
- Example of available information when finding out data:
- population mean (65")
- population standard deviation (3.5")
- sample size (N = 35)
- sample mean (67")
- Conduct a one-sample z test in this case
- Choose whether to conduct a one-directional or a non-directional test.
Step 5: Characterize the sampling distribution
- Take all possible random samples of size 35, compute means, plot them
- To compute the mean and SEM of the sampling distribution:
- σx = σ/√N = 3.5/√35 = 3.5/5.916 = 0.592
- After this mark Zcrit on to the sampling distribution, where Zcrit=1.645
Step 6: Compute the Test Statistic
- Zobs is MORE EXTREME compared to Zcrit therefore we reject the null hypothesis.
- Zobs falls in the REJECTION REGION of the sampling distribution
- Formula for the test statistic (Zobs) as well as critical value of Z:
- Zobs = (X - μ) / σx = (67-65) /0.592 = 3.38
Step 7: Make a Decision about H0
- Conclude that it is unlikely that the sample mean of 67" came from a population with a mean of 65".
Why can't we "accept" the Null Hypothesis
- It is possible that a sample mean of 67" could have come from this population.
- It is not possible to prove the null
- It is highly unlikely since we can conclude we "fail to reject the null" NOT we "accept the null" considering we can never prove a null hypothesis true; we can only disprove it.
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Description
Understand hypothesis testing with z-tests and t-tests. Learn about effect size (Cohen's d), Type I and II errors, and statistical power. Review sample statistics, population parameters, and the central limit theorem.
