Hypothesis testing Flashcards
(28 cards)
what is a Significance Level ?
The smallest probability that an event could occur by chance
What is the difference between a one-tailed and two-tailed hypothesis test ?
One-tailed test: Tests if the population parameter has either increased or decreased (H1: θ >)
Two-tailed test: Tests if the population parameter has changed in any direction (H1: θ ≠)
What is a test statistic in hypothesis testing ?
A numerical value calculated from sample data used to determine whether to accept or reject the null hypothesis.
What does the significance level represent in a hypothesis test ?
The probability threshold for deciding if the observed data is unlikely to occur by chance, typically set before the test
How do we decide whether to reject or accept the null hypothesis ?
Reject the null hypothesis if the p-value is less than the significance level or if the test statistic falls within the critical region.
p-value: The probability of observing a value at least as extreme as the test statistic, assuming the null hypothesis is true.
what do you do to the p value in two-tailed tests ?
double it or half the significance level
How do we calculate the p-value for a one-tailed test ?
If the test looks for a decrease, find the probability of the test statistic being less than or equal to the observed value.
If the test looks for an increase, find the probability of the test statistic being greater than or equal to the observed value.
What is a critical region in hypothesis testing ?
The critical region is the range of test statistic values that lead to the rejection of the null hypothesis.
What is the critical value ?
The critical value is the boundary of the critical region.
It is the least extreme value that would result in rejecting the null hypothesis, determined by the significance level.
What is the actual significance level in hypothesis testing ?
The actual significance level is the probability of incorrectly rejecting the null hypothesis
What are the steps for carrying out a hypothesis test ?
Define the test statistic and population parameter.
Write the null hypothesis (H₀) and alternative hypothesis (H₁).
Calculate the critical value(s) or the p-value for the test.
Compare the observed test statistic with the critical value(s) or compare the p-value with the significance level.
Decide whether there is enough evidence to reject H₀.
Write a conclusion in the context of the problem.
How should a conclusion be written for a hypothesis test ?
Rejecting H₀: State that there is sufficient evidence to suggest the alternative hypothesis is true at the given significance level.
Accepting H₀: State that there is not enough evidence to suggest the alternative hypothesis is true at the given significance level.
What should be included in the conclusion of a two-tailed hypothesis test?
The conclusion should state whether there is evidence of a change in the population parameter
What is the population parameter tested in a binomial distribution hypothesis test ?
The population parameter tested is the probability p in a binomial distribution B(n, p), where n is the number of trials.
How do I write the null and alternative hypotheses for a hypothesis test using the binomial distribution ?
Null hypothesis:
𝐻0 : p = [assumed value of p]
Alternative hypothesis:
One-tailed test:
𝐻1 : 𝑝 > [value] or
𝐻1 : 𝑝<H1[value]
Two-tailed test:
𝐻1 : 𝑝 ≠ [value]
What is the test statistic used in a hypothesis test with the binomial distribution ?
The number of successes observed in the given number of trials, n.
How do I calculate the p-value or critical region in a binomial distribution hypothesis test ?
p-value: Calculate the probability of the observed test statistic or a more extreme value. Compare this with the significance level.
Critical region: Find the critical value(s) using your calculator by determining when the probability of observing the test statistic is less than the significance level.
For one-tailed tests, check for p ≤ α or p ≥ α based on the direction of the test.
For two-tailed tests, use half the significance level for both tails.
What are the steps to follow when carrying out a hypothesis test with the binomial distribution ?
Define the probability, p.
Write the null and alternative hypotheses.
Define the test statistic (number of successes).
Calculate the p-value or critical value(s).
Compare the test statistic with the critical value(s) or p-value with the significance level.
Decide if there is enough evidence to reject the null hypothesis.
Write a conclusion in context.
What is the distribution of the sample means ?
The distribution of the sample means is the distribution of all possible values that the sample mean (𝑋‾) can take when sampling from a population.
What is the mean and variance of the distribution of sample means ?
Mean of sample means:
𝜇 (same as the population mean)
Variance of sample means:
𝜎^2/𝑛
Standard deviation of sample means:
𝜎/root 𝑛
What happens to the variance and standard deviation of the sample means as the sample size,
𝑛 increases ?
Variance decreases because it is inversely proportional to 𝑛
Standard deviation also decreases, becoming narrower as sample size increases.
How does the sample size (n) affect the distribution of sample means ?
The greater the sample size 𝑛, the less the sample means will vary, meaning the distribution becomes narrower. The standard deviation decreases with the square root of the sample size, making the distribution of sample means more concentrated around the population mean.
What is the population parameter being tested in a hypothesis test using the normal distribution ?
The population parameter being tested is the population mean, 𝜇, in a normally distributed random variable
X∼N (μ , σ^2).
What are the null and alternative hypotheses in a hypothesis test for the mean of a normal distribution ?
Null hypothesis (H0):
𝜇 = 𝜇0
(The population mean is equal to a specific value 𝜇0)
Alternative hypothesis (H1):
One-tailed test:
𝜇 > 𝜇0
or
𝜇 < 𝜇0
Two-tailed test:
𝜇 ≠ 𝜇0
(The population mean has changed)
