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Hypothesis Testing
AP Statistics - Inference for Quantitative Data: Means
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Terms in this set (25)
Conditions for Inference About a Population Mean
Random sample - the data are a random sample from the population of interest
10% rule - the sample size is no more than 10% of the population size
Large counts - the population has a Normal distribution or the sample size is large (n ≥ 30)
Central Limit Theorem
the theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution
If the population distribution has unknown shape and n < 30, what do we use?
use a graph of the sample data to assess the Normality of the population: dot plot, histogram, box plot or normal probability plot. Do not use t procedures if the graph shows strong skewness or outliers
Standard error of a sample mean
One-sample T-interval
or use TInterval and specify the degrees of freedom
Sample size for a desired margin of error
To determine the sample size (n) for a given margin of error (MOE) in a 1 sample t interval, solve for in the equation
MOE = z* (s/square root of n)
Conditions for Performing Inference About
mean1 - mean2
Random: The data come from two independent random samples or from two groups in a randomized experiment
10%: When sampling without replacement, check that n1 ≤ (1/10)N1 and n2 ≤ (1/10)N2.
Normal/Large Sample: Both population distributions (or the true distributions of responses to the two
treatments) are Normal or both sample sizes are large (n1 ≥ 30 and n2 ≥ 30 or a data plot shows no strong skewness or outliers.
When the conditions are met, an approximate C% confidence interval for μ1 - μ2 is:
2SampTInterval and specify degrees of freedom
How do you find degrees of freedom?
n-1
Step 1 for One Sample t-test for a Population Mean
Hypothesis: H0: μ = μ0; Ha: μ < μ0 or μ > μ0 or μ ≠ μ0 where μ is ...
Step 2 for One Sample t-test for a Population Mean
Conditions:
Random: The data come from a well-designed random sample or randomized experiment.
Independent: 10%: When sampling without replacement, check that n ≤ (1/10)N.
Normal: Large Sample: The population has a Normal distribution or the sample size is large (n ≥ 30).
If the population distribution has unknown shape and n < 30, use a graph of the sample data to assess the Normality of the population. Do not use t procedures if the graph shows strong skewness or outliers.
Step 3 for One Sample t-test for a Population Mean
Test Statistic
t = (xbar - null)/standardx/sq.root(n)
p-value
use T-Test
Step 4 for One Sample t-test for a Population Mean
Conclusion: If P < a, then Reject the H0, otherwise Fail to Reject H0. Answer in context.
Paired Differences T-Test
To compare the responses to the two treatments in a paired data design, apply the one-sample t procedures to the observed differences (mean difference)
Conditions for Significance Test for a Difference in Means
Random - We have two random samples, from two distinct populations
Independence - Each sample must be selected independently of the other (no pairing or matching) and each distinct population size must be 10 times greater than their samples
Normality - Counts of each sample are at least 30 or use graphs of each sample to assess Normality.
Finding the significance test for a difference in means
2SampT-Test
state the degrees of freedom in your calculator
Type I error
rejecting a true null hypothesis
Type II error
failing to reject a false null hypothesis
The probability of a Type I error is the same as...
alpha, the significance level
The probability of a Type II error is...
beta, or 1 - power
The power of a test is...
the probability that the test will reject Ho at a chosen significance level when the specified
alternative value of the parameter is true
Power is the probability of...
avoiding a Type II error
Decreasing the probability of a Type II error is...
the exact same thing as increasing power
How do we increase power of a significance test?
Increase sample size. A larger sample size gives more information about the true parameter.
Increase the significance level. Using a larger significance level makes it easier to reject H0 when Ha is true.
Increase difference between the null and alternative parameter values (effect size).
If it is more important to avoid a Type I error, choose a _____ a. If it is more important to avoid a Type II error, choose a
_______ a.
smaller
larger
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