Stats Exam III
What is the critical value? For a two-tail test: The critical values occur in both the upper and lower tails of the standard normal distribution.
With a level of significance of =.05, the area in each tail beyond the critical values = .025. • We find the z-scores that correspond to these areas by looking them up in the table. • The z -score that corresponds to an area of 025 in the lower tail = -1.96. • The z -score that corresponds to an area of 1-025 = .975 in the upper tail = +1.96. • So the critical value = ±1.96.
hypothesis testing
a process that uses sample data to evaluate the probability of a hypothesis about a population parameter
point estimate
a single value representing the best estimate of the population parameter - ex: x bar is u - ex: x bar - u(null) when doing u1 - u2
Which is more powerful? Dependent t-tests or independent means t-tests?
dependent means t-test because the observations are made on the same subjects twice )or matched pairs of subjects) they will be correlated. Usually this correlation will be positive.
interval estimation
determines a range of values that define an interval where the population parameter is likely to fall.
if the confidence interval included the parameter (u value at beg), then we
do not reject the null
statistical hypothesis
more formally structured hypotheses that are based on research hypotheses including null and alternative hypothesis
one-tailed tests are
more powerful than two tailed tests because as alpha level increases, critical value decreases (easier to reject) b/c region of rejection increases.
statistically significant
not likely to have occurred by chance alone/sampling error
directional alternative hypothesis uses a
one-tailed test and region of rejection is only in 1 tail, so magnitude of the effect will not be as large as for non-directional in order to reject the null
What is the probability of observing a sample mean 3 standard errors from (above or below) the mean of the population?
p (z ≤ 3) + p (z ≤ -3) (this is the area included in both tails). 1 - .9987 = .0013 (this is the area included in the tail above z). p (z ≤ -3) = .0013 (this is the area included in the tail above z). .0013 + .0013 = .0026
if the confidence interval does not include the parameter, then we
reject the null
Type I error
reject the null hypothesis, but in reality, the null is true and should be retained (we should fail to reject) - the probability of making a Type I Error is equal to our elected alpha level - we have stated that a significant effect does exist in the population when it does not and we have found it by chance alone due to sampling error
To decrease the risk of making a Type I Error, we can
select a smaller value of alpha - the smaller the value of alpha, the larger effect must be to be considered statistically significant - round down because this increases critical value, which makes null harder to reject
standard error of the difference b/t means
standard deviation of the sampling distribution of the diff. b/t the means - tells us how much the diff. b/t the sample means varies around (deviates from) the diff. b/t pop. means
Sdbar
standard error of the mean of difference scores
n>30
t-dis. approximates a normal dis.
n = infinity
t-dis. is normal
null hypothesis
the hypothesis we test; always assumed to be true in the population. always speculates no/not; status quo - always uses some form of =, < or = to, > or = to
D0 =
the hypothesized difference between u1 and u2; value is often 0 and is therefore eliminated from the hypotheses/equations.
d bar =
the mean/average of the difference scores
critical values
the numerical values in the distribution that define the boundaries of the critical region (region of rejection) Ex: when alpha2 = .05, critical values = + or - 1.96
Power of a statistical test
the probability of correctly rejecting the null hypothesis: 1- beta
p =
the probability of getting the observed value of our test statistic if the null is true (results occurred by chance alone)
For an upper tail test, the p-value (also called the observed level of significance) is
the probability of obtaining a value that is equal to or greater than the observed test statistic, if the null hypothesis is true.
alpha =
the probability of rejecting the null when in fact it is true
Sd =
the standard deviance of the difference scores
independent means test
used to compare the means of 2 mutually exclusive groups.
Dependent Means t-test
used to determine whether there is a statistically significant difference b/t the means of 2 related groups - related when 2 observations are made on the same subjects
n>120
very close to a normal dis.
when standard deviation is known...
we can use z-scores to test hypotheses about means and proportions
when standard deviation is not known,
we use the t-distribution to test hypotheses about means and proportions - more likely to produce a type I error if you use z
research hypothesis
what we believe will happen - the researcher's belief about the outcome of an experiment, study, or intervention - based on theory, research, or prior experience
region of rejection
within our sampling distribution, the area that represents the values of the sample statistic that are so extreme, that they are not probable/not likely of the null is true
d =
x1-x2
critical values are lower for directional hypotheses than they are for non-directional b/c...
you multiply non-directional (2 tailed) z values by 2 - region of rejection is only in 1 tail, so magnitude of the effect will not be as large as for non-directional in order to reject the null
hypothesis for dependent means t-test
➢H0 :u1 = u2 ➢H1 :u1 ≠ u2 ➢H1 :u1 > u2 orH1 :u1 < m2 ➢H0 :uD = 0 ➢H1 :uD ≠ 0 ➢H1 :uD > 0 or H1 :uD < 0
t-distribution
- A symmetric bell-shaped distribution, shape varies based on sample size - smaller the sample size, the flatter the distribution - The larger the sample size gets, the closer our sample estimate of the standard deviation of the population gets to the parameter itself, and the more closely the t distribution resembles the normal (z) distribution. - as sample size increases, degrees of freedom increase
dependent means t-test limitations
- The increase might be due to other factors. In this particular case - in-school instruction. - Two groups should have been examined: a group of students who did participate in the after school program and a group of students who did not participate in the after school program. - we cannot say that the after school program is effective
one-sample t-tests compared to one-sample z-tests
- a sample mean is compared to some fixed value that represents a pop. mean - for both, the diff. b/t a sample mean and pop mean is divided by sampling error
Type III Error
- a type of error possible with one-tailed tests in which a decision would have been to reject the null hypothesis, but the researcher decides to retain the null hypothesis because the rejection region was located in the wrong tail - samples mean falls opposite f hypothesized direction in alternative
sampling distribution of the difference between means
- if diff. b/t means is so larger that it couldn't be reasonably be accounted for by chance variation when pop means are the same, we reject null
directional alternative hypothesis
- one tailed test - if > then upper tail and < lower tail - always refers to population parameters, never sample statistics
level of significance
- the higher the value of alpha, the higher the power - a larger alpha makes it easier to reject null bc larger level, more area is located in region of rejection - higher the alpha level, the lower the critical value for a test stat
sample size
- the larger the sample size, the higher the power - as the size increases, the sample stat becomes a better estimator of the pop. parameter - easiest way to increase stat. power is by inc. sample size - as n increases, the value of the standard error decreases and the observed value increases = more powerful
variability in the population
- the lower the variability, the higher the power - as the standard deviation of the population becomes smaller, the standard error gets smaller, and the observed value of the test sta. increases, making the test stat. more powerful
Effect size
- the magnitude of a relationship between two or more variables - easier to detect larger effect sizes (larger differences or relationships) than smaller ones - the higher the effect size, the higher the power - test stat = effect/error - effect increases, observed value increases
Type II Error
- when the null hypothesis is actually false, but we fail to reject it. - probability is called beta - we have stated that a significant effect does not exist in the population when it does
one-sample t-tests compared to one-sample z-tests (differences)
- z-test: sampling error is measured by standard error of the mean - t-test, because the pop standard deviation is unknown, the standard error of the mean is estimated, by dividing the estimated standard deviation by square root of sample size
differences between the t-distribution and the z-distribution
-greater area under the t tails in t distribution compared to normal dis. - t scores are based on size and variability, so t-scores critical are higher than critical z scores (especially for small samples - there is a different t-distribution for each sample size
sampling distribution of the difference b/t means is a
-prob. distribution - describes what diff. (b/t means) would exist if the null were true and the 2 pop. means were the same
the probability of correctly rejecting a false null hypothesis (not making a a type II error) =
1- beta - called the power of the statistical test - want to have a high power; high probability of correctly rejecting a false null
Statistical hypothesis with independent means test
H0 : m1 - m2 = D0 H1 : m1 - m2 ≠ D0 H0 : m1 - m2 ≤ D0 H1 : m1 - m2 > D0 H0 : m1 - m 2 ≥D0 H1 : m1 - m 2 < D0
Interpret your confidence interval with respect to the research question p = .64 95% CI = .42 ≤ p ≤ .62
We conclude that the percentage of supermarket shoppers who believe that the supermarket ketchup is as good as national name-brand ketchup differs significantly from 64%.
sampling dis. of the diff. b/t means properties
1. Its mean is equal to 0: the positive values of x bar 1 - x bar 2 cancel out the negative values of x bar 1 - x bar 2. 2. Therefore, the mean of this distribution is equal to the true difference between population means for H0 : u1 -u2 =0. 3. For each sample, the sampling distribution of the mean will approach a normal distribution as n increases. At n>30, it is basically normal: this is the Central Limit Theorem. 4. If the sampling distributions of the mean for each sample are approximately normal, then the distribution of the difference between pairs of means will also be approximately normal. 5. The standard deviation of the sampling distribution of the difference between means is called the standard error of the difference between means
one-sample t-test assumptions
1. cases are randomly sampled from the population 2. data have been sampled from a normally distributed population
Power of a statistical test is influenced by 4 factors:
1. level of significance (alpha) 2. sample size 3. effect size 4. variability in the population
we can reject the null when...
1. when p is less then or equal to alpha 2. when the abs value of the test stat. observed is greater than or equal to abs value of test stat. critical
Non-directional hypothesis uses a
2 tailed test and region of rejection is located in both tails
matched pairs design
A method of assigning subjects to groups in which pairs of subjects are first matched on some characteristic and then individually assigned randomly to groups.
Non-directional hypothesis
A two-tailed hypothesis that does not predict the direction of the difference or relationship (e.g. girls and boys are different in terms of helpfulness). - uses not equal to for alternative
conclusion of dep. means t-test
There is a significant difference between the mean of the pre-test and the mean of the post-test. Students who participated in the after school program increased their scores.
interpret results for dependent means test
There is not a statistically significant difference in the average P/E (price to earnings) ratio for companies from one year to the next.
If a hypothesis is rejected at 95% confidence, it a. will always be accepted at 90% confidence b. will always be rejected at 90% confidence c. will sometimes be rejected at 90% confidence d. None of these alternatives is correct.
b - The critical value is higher and the probability of rejection is smaller for .05 than it is for .10.
Direct diff. method
calculate the difference scores (d) for each pair of scores first, and use the difference scores (d) as our unit analysis
the standard error of the diff. b/t independent means...
combines the variance from each independent population into one measure
alternative hypothesis
contradicts the null hypothesis; usually presents the outcome we believe to exist in the population. - null hypothesis and alternative hypothesis are mutually exclusive (only one can be true)
critical prob vs critical value vs p-value
critical prob = alpha critical value = find alpha on z table p-value = use observed value