Exam 2 Terms Flashcards
What does a hypothesis test do?
A hypothesis test uses data from a sample to assess a claim about a population.
When do we not need a hypothesis test?
If we have data for the entire population.
Describe the null hypothesis.
- Contains an equality ( = )
- is in terms of parameters, not statistics
Describe the alternative hypothesis.
- sign can be not equal to, > , or <
- in terms of parameters, not statistics
What does it mean in the null hypothesis if rho = 0?
It means there is no correlation
rho represents a population correlation
What is the p-value?
The probability of obtaining results (in direction of the alternative hypothesis) as extreme as, or more extreme than, those observed if H0 is true.
If our p-value is small enough, then we have convincing evidence against H0 in favor of Ha.
p-value < alpha
reject the H0, results are statistically significant
p-value > alpha
fail to reject the H0, results are not statistically significant
Is failing to reject the null hypothesis the same as accepting it?
no, we never say we “accept” H0
What are the steps of testing a hypothesis?
- State H0 and Ha
- calculate sample statistics
- calculate the test statistic
- acquire p-value
- make a decision about H0 and Ha based on alpha
- write conclusion
What does t-distribution shape depend on?
sample size where degrees of freedom = n-1
What is the formula for test statistic?
t = x bar - mu / (s/sqrt n)
sample mean minus hypothesized population mean divided by the sample standard deviation over the square root of the sample size
What is standard error?
s/ sqrt n
How do we determine what “tails” to use when calculating p-value?
Equal tails in Ha does not equal whatever you’re testing (must multiply p-value by 2)
right tail if Ha > than what you’re testing
left tail if Ha < than what you’re testing
What assumptions must be met to use a t-test?
Sample size must be greater than or equal to 30
Data should be bell shaped with no extreme outliers
What p-value provides greater evidence against H0?
the smaller p-value
ex p = 0.0031 provides stronger evidence against H0 than p = 0.0032
What is alpha?
the significance level or probability of making a Type I error
can be assumed to be 95% (0.05) if not stated
If the sample mean is less than the hypothesized mean, will the t test statistic be negative or positive?
negative
If the sample mean is greater than the hypothesized mean, will the t test statistic be negative or positive?
positive
What happens to the test statistic as the difference between the sample mean and the hypothesized mean increases?
What happens to the p-value in this scenario?
Test statistic gets farther away from 0
The p-value gets smaller
What happens to the p-value if the test statistic is close to 0?
It gets larger
What happens if the sample mean equals the hypothesis mean in a t test?
the test statistic would be 0, and p = 1, so there is no evidence to reject the null
What is multiple testing?
Suppose H0 is true.
When multiple studies are conducted, alpha (ex. 5%) of all p-values will be statistically significant just by random chance.
example:
Given we took 20 samples and computed 20 p-values, how many p-values would we expect to be significant, even though H0 is actually true given alpha = 0.05?
20 * 0.05 = 1
5% of people (1 out of the 20) would incorrectly reject the null.
What makes multiple testing worse?
Publication bias
Often, only significant results are published. If many tests are conducted, some of them will be statistically significant just by chance, and we may only hear about those significant studies.
What are the parts of a confidence interval?
CI ( __ , __)
x bar plus or minus t* x s/sqrt n
x bar is the point estimate or best estimate
t* x s/sqrt n is the margin of error
What is 1 - alpha?
The confidence interval
ex. is alpha is 0.05, the CI is 95%
What should t* be close to usually?
2
How do we interpret confidence intervals?
Essentially means that we are ___ % confident that the population mean (or parameter) for whatever we are testing is between ( __ , __ )
100% sure that the sample mean is in that range since it was used to calculate CI
Ex. We are 95% confident that the population mean pH for all Florida lakes is between 6.236 and 6.946.
What is the point estimate?
The statistic that serves as the best estimate for the parameter.
X bar or the mean of the sample
What is the margin of error?
Half of the width of a confidence interval; equal to t* times the standard error.
t* x s/sqrt n
Given a CI, how do we find the point estimate or the margin of error?
To find point estimate, find the average of the confidence interval. (i.e. add the two values in the parentheses and divide by 2)
To find ME, subtract the two values in the CI and divide by 2
What happens to t* as CI increases
t* increases
Which is wider, a 99% CI or a 95% CI?
99%
How do we solve for sample size for a mean?
n = ((t* or z*)(s)/ME)^2
ALWAYS ROUND UP FOR SAMPLE SIZE
How can we reject or fail to reject a null hypothesis based on the CI?
If the value indicated within the null hypothesis falls within the confidence interval, we fail to reject it. Reject the null only if it is not within the CI.
What is z* for a 95% CI?
1.96
What is z* for a 90% CI?
1.645
What is z* for a 99% CI?
2.576
What happens to n as the variability decreases?
n decreases (s gets smaller, so numerator gets smaller)
What happens to n as ME decreases?
n increases (ME is in denominator, smaller ME means larger n)
How do we write hypotheses for a two samples?
mu of one group = mu of other group
How do we calculate a two sample t test?
t = (xbar 1 -xbar 2)/ sqrt (s1^2/n1 + s2^2/n2)
For a two sample t test, how do we find degrees of freedom?
df = n - 1 for the smaller n
Compare t-test statistic to t*
We use t-test statistic to calculate p-value
We use t* to calculate CI
How do we calculate CI for two means?
(x1 - x2) plus or minus t* x sqrt (s1^2/n1 + s2^2/n2)
How do we interpret CI for two means?
We are __ % confident that the difference in population means is between ( __, __ )
How can we reject the null based on a CI for a two sample test?
If 0 falls within the CI, fail to reject the null
can reject null is 0 is not within CI
What are independent samples?
From a randomized comparative experiment. Each case is randomly assigned to only one of the treatment conditions.
Cases in each group are unrelated to one another
What are paired samples?
From a matched pair experiment. Each case is measured under BOTH treatment conditions.
Cases in each group are meaningfully matched with one another; also known as dependent samples or matched pairs
How do we write hypotheses tests for a case of differences?
mu will always equal 0
How do we calculate t test statistic for a sample of differences?
the same as for a one sample t test, but use the mean of the differences and the stdev of a difference
What is a sampling distribution?
The distribution for a statistic. It shows how a statistic varies from sample to sample.
How can we describe standard error in terms of sampling distribution?
The standard error is the standard deviation of the sampling distribution.
